Optical profilometry of additional-material deviations in a periodic grating

ABSTRACT

Disclosed is a method and system for measurement of periodic gratings which have deviations which result in more than two materials occurring along at least one line in the periodic direction. A periodic grating is divided into a plurality of hypothetical layers, each hypothetical layer having a normal vector orthogonal to the direction of periodicity, each hypothetical layer having a single material within any line parallel to the normal vector, and at least one of the hypothetical layers having at least three materials along a line in the direction of periodicity. A harmonic expansion of the permittivity ∈ or inverse permittivity 1/∈ is performed along the direction of periodicity for each of the layers including the layer which includes the first, second and third materials. Fourier space electromagnetic equations are then set up in each of the layers using the harmonic expansion of the permittivity ∈ or inverse permittivity 1/∈, and Fourier components of electric and magnetic fields in each layer. The Fourier space electromagnetic equations are then coupled based on boundary conditions between the layers, and solved to provide the calculated diffraction spectrum.

RELATED DOCUMENTS

[0001] The present application is based on Disclosure Document serialnumber 474051, filed May 15, 2000, entitled Optical Profilometry forPeriodic Gratings with Three or More Materials per Layer by the sameinventors. Therefore, it is requested that the above-specifiedDisclosure Document be retained in the file of the present patentapplication.

TECHNICAL FIELD

[0002] The present invention relates generally to the measurement ofperiodic surface profiles using optical techniques such as spectroscopicellipsometry. In particular, the present invention relates to opticalprofilometry of profile deviations of semiconductor fabricationprocesses, more particularly, additional-material deviations in aperiodic grating.

BACKGROUND OF THE INVENTION

[0003] There is continual pressure on the semiconductor microchipindustry to reduce the dimensions of semiconductor devices. Reduction inthe size of semiconductor chips has been achieved by continuallyreducing the dimensions of transistors and other devices implemented onmicrochip arrays. As the scale of semiconductor devices decreases,control of the complete profile of the features is crucial for effectivechip operation. However, limitations in current fabrication technologiesmake formation of precise structures difficult. For example, completelyvertical sidewalls and completely horizontal top and bottom surfaces indevice formation are difficult, if not impossible, to achieve. Slopingsidewalls and top and bottom surfaces are common. Additionally, otherartifacts such as “T-topping” (the formation of a “T” shaped profile)and “footing” (the formation of an inverse “T” shaped profile) arecommon in microchip manufacturing. Metrology of such details about theprofile is important in achieving a better understanding of thefabrication technologies. In addition to measuring such features,controlling them is also important in this highly competitivemarketplace. There are thus increasing efforts to develop and refinerun-to-run and real-time fabrication control schemes that includeprofile measurements to reduce process variability.

[0004] Optical metrology methods require a periodic structure foranalysis. Some semiconductor devices, such as memory arrays, areperiodic. However, generally a periodic test structure will befabricated at a convenient location on the chip for optical metrology.Optical metrology of test periodic structures has the potential toprovide accurate, high-throughput, non-destructive means of profilemetrology using suitably modified existing optical metrology tools andoff-line processing tools. Two such optical analysis methods includereflectance metrology and spectroscopic ellipsometry.

[0005] In reflectance metrology, an unpolarized or polarized beam ofbroadband light is directed towards a sample, and the reflected light iscollected. The reflectance can either be measured as an absolute value,or relative value when normalized to some reflectance standard. Thereflectance signal is then analyzed to determine the thicknesses and/oroptical constants of the film or films. There are numerous examples ofreflectance metrology. For example, U.S. Pat. No. 5,835,225 given toThakur et. al. teaches the use of reflectance metrology to monitor thethickness and refractive indices of a film.

[0006] The use of ellipsometry for the measurement of the thickness offilms is well-known (see, for instance, R. M. A. Azzam and N. M.Bashara, “Ellipsometry and Polarized Light”, North Holland, 1987). Whenordinary, i.e., non-polarized, white light is sent through a polarizer,it emerges as linearly polarized light with its electric field vectoraligned with an axis of the polarizer. Linearly polarized light can bedefined by two vectors, i.e., the vectors parallel and perpendicular tothe plane of incidence. Ellipsometry is based on the change inpolarization that occurs when a beam of polarized light is reflectedfrom a medium. The change in polarization consists of two parts: a phasechange and an amplitude change. The change in polarization is differentfor the portion of the incident radiation with the electric vectoroscillating in the plane of incidence, and the portion of the incidentradiation with the electric vector oscillating perpendicular to theplane of incidence. Ellipsometry measures the results of these twochanges which are conveniently represented by an angle Δ, which is thechange in phase of the reflected beam ρ from the incident beam; and anangle Ψ, which is defined as the arctangent of the amplitude ratio ofthe incident and reflected beam, i.e.,${\rho = {\frac{r_{p}}{r_{s}} = {{\tan (\Psi)}^{j\quad {(\Delta)}}}}},$

[0007] where r_(p) is the p-component of the reflectance, and r_(s) isthe s-component of the reflectance. The angle of incidence andreflection are equal, but opposite in sign, to each other and may bechosen for convenience. Since the reflected beam is fixed in positionrelative to the incident beam, ellipsometry is an attractive techniquefor in-situ control of processes which take place in a chamber.

[0008] For example, U.S. Pat. No. 5,739,909 by Blayo et. al. teaches amethod for using spectroscopic ellipsometry to measure linewidths bydirecting an incident beam of polarized light at a periodic structure. Adiffracted beam is detected and its intensity and polarization aredetermined at one or more wavelengths. This is then compared with eitherpre-computed libraries of signals or to experimental data to extractlinewidth information. While this is a non-destructive test, it does notprovide profile information, but yields only a single number tocharacterize the quality of the fabrication process of the periodicstructure. Another method for characterizing features of a patternedmaterial is disclosed in U.S. Pat. No. 5,607,800 by D. H. Ziger.According to this method, the intensity, but not the phase, ofzeroth-order diffraction is monitored for a number of wavelengths, andcorrelated with features of the patterned material.

[0009] In order for these optical methods to be useful for extraction ofdetailed semiconductor profile information, there must be a way totheoretically generate the diffraction spectrum for a periodic grating.The general problem of electromagnetic diffraction from gratings hasbeen addressed in various ways. One such method, referred to as“rigorous coupled-wave analysis” (“RCWA”) has been proposed by Moharamand Gaylord. (See M. G. Moharao and T. K. Gaylord, “RigorousCoupled-Wave Analysis of Planar-Grating Diffraction”, J. Opt. Soc. Am.,vol. 71, 811-818, July 1981; M. G. Moharam, E. B. Grann, D. A. Pommetand T. K. Gaylord, “Formulation for Stable and Efficient Implementationof the Rigorous Coupled-Wave Analysis of Binary Gratings”, J. Opt. Soc.Am. A, vol. 12, 1068-1076, May 1995; and M. G. Moharam, D. A. Pommet, E.B. Grann and T. K. Gaylord, “Stable Implementation of the RigorousCoupled-Wave Analysis for Surface-Relief Dielectric Gratings: EnhancedTransmittance Matrix Approach”, J. Opt. Soc. Am. A, vol. 12, 1077-1086,May 1995.) RCWA is a non-iterative, deterministic technique that uses astate-variable method for determining a numerical solution. Severalsimilar methods have also been proposed in the last decade. (See P.Lalanne and G. M. Morris, “Highly Improved Convergence of theCoupled-Wave Method for TM Polarization”, J. Opt. Soc. Am. A, 779-784,1996; L. Li and C. Haggans, “Convergence of the coupled-wave method formetallic lamelar diffraction gratings”, J. Opt. Soc. Am. A, 1184-1189,June, 1993; G. Granet and B. Guizal, “Efficient Implementation of theCoupled-Wave Method for Metallic Lamelar Gratings in TM Polarization”,J. Opt. Soc. Am. A, 1019-1023, May, 1996; U.S. Pat. No. 5,164,790 byMcNeil, et al; U.S. Pat. No. 5,867,276 by McNeil, et al; U.S. Pat. No.5,963,329 by Conrad, et al; and U.S. Pat. No. 5,739,909 by Blayo et al.)

[0010] Generally, an RCWA computation consists of four steps:

[0011] The grating is divided into a number of thin, planar layers, andthe section of the ridge within each layer is approximated by arectangular slab.

[0012] Within the grating, Fourier expansions of the electric field,magnetic field, and permittivity leads to a system of differentialequations for each layer and each harmonic order.

[0013] Boundary conditions are applied for the electric and magneticfields at the layer boundaries to provide a system of equations.

[0014] Solution of the system of equations provides the diffractedreflectivity from the grating for each harmonic order.

[0015] The accuracy of the computation and the time required for thecomputation depend on the number of layers into which the grating isdivided and the number of orders used in the Fourier expansion.

[0016] The diffracted reflectivity information which results from anRCWA computation can be used to determine the details of the profile ofa semiconductor device. Generally, reflectivities for a range ofdifferent possible profiles of a given semiconductor device arenumerically calculated using RCWA and stored in a database library.Then, the actual diffracted reflectivity of the given device is measuredas disclosed, for example, in co-pending U.S. patent application Ser.No. 09/764,780 for Caching of Intra-Layer Calculations for RapidRigorous Coupled-Wave Analyses filed Jan. 25, 2000 by the presentinventors which is hereby incorporated in its entirety into the presentspecification, or X. Niu, N. Jakatdar, J. Bao and C. J. Spanos,“Specular Spectroscopic Scatterometry” IEEE Trans. on SemiconductorManuf., vol. 14, no. 2, May 2001. The reflected phase and magnitudesignals obtained, in the case of ellipsometry, and relative reflectance,in the case of reflectometry, are then compared to the library ofprofile-spectra pairs generated stored in the library. A phase and/oramplitude measurement will be referred to in the present specificationas the “diffracted reflectivity.” The matching algorithms that can beused for this purpose range from simple least squares approach, to aneural network approach that associates features of the signal with theprofile through a non-linear relationship, to a principalcomponent-based regression scheme. Explanations of each of these methodsis explained in numerous text books on these topics such as Chapter 14of “Mathematical Statistics and Data Analysis” by John Rice, DuxburyPress and Chapter 4 of “Neural Networks for Pattern Recognition” byChristopher Bishop, Oxford University Press. The profile associated withthe RCWA-generated diffracted reflectivity that most closely matches themeasured diffracted reflectivity is determined to be the profile of themeasured semiconductor device.

[0017] In semiconductor manufacturing, a number of processes may be usedto produce a periodic structure having two materials in the periodicdirection. In the present specification the “nominal” number ofmaterials occurring in the periodic direction is considered to be themaximum number of materials that lie along any of the lines which passthrough the periodic structure in the direction of the periodicity.Accordingly, structures having a nominal two materials in the periodicdirection have at least one line along the direction of periodicitypassing through two materials, and no lines along the direction ofperiodicity passing through more than two materials. Additionally, itshould be noted that when specifying the nominal number of materialsoccurring along a periodic direction of a structure in the presentspecification, the gas, gases or vacuum in gaps between solid materialsis considered to be one of the materials. For instance, it is notnecessary that both materials occurring in the periodic direction of anominal two-material periodic structure be solids.

[0018] An example of a structure 100 with two materials in a layer isshown in the cross-sectional view of FIG. 1A, which shows two periods oflength D of a periodic portion of the structure 100. The structure 100consists of a substrate 105, with a thin film 110 deposited thereon, anda periodic structure 120 on the film 110 which consists of a series ofridges 121 and grooves 122. In exemplary structure 100, each ridge 121has a lower portion 131, a middle portion 132 and an upper portion 133.It should be noted that according to the terminology of the presentinvention, the lower, middle and upper portions 131-133 are not‘layers.’ In exemplary structure 100 of FIG. 1A, the lower, middle andupper portions 131-133 are each composed of a different material. Thedirection of periodicity is horizontal on the page of FIG. 1A, and itcan be seen that a line parallel to the direction of periodicity maypass through at most two different materials. For instance, a horizontalline passing through the middle portion 132 of one of the ridgestructures 121, passes through the middle portion 132 of all of theridge structures 121, and also passes through the atmospheric material122. That is, there are two materials in that region. (It should benoted that a line which is vertical on the page of FIG. 1A can passthrough more than two materials, such as a line passing through thelower, middle and upper portions 131-133 of a ridge structure, the thinfilm 110, and the substrate 105, but according to the terminology of thepresent specification this structure 100 is not considered to have anominal three or more materials in the periodic direction.)

[0019] A close-up cross-sectional view of a ridge structure 121 is shownin FIG. 1B with the structure being sectioned into what are termed‘harmonic expansion layers’ or simply ‘layers’ in the presentspecification. In particular, the upper portion 133 is sectioned intosix harmonic expansion layers 133.1 through 133.5, the middle portion132 is sectioned into nine harmonic expansion layers 132.1 through132.9, the lower portion 131 is sectioned into six harmonic expansionlayers 131.1 through 131.6, and five harmonic expansion layers 110.1through 110.5 of the thin film 110 are shown. All layer boundaries arehorizontal planes, and it should be understood that harmonic expansionlayers 133.1-133.5, 132.1-132.9, 131.1-131.6 and 110.1-110.5 may havediffering thicknesses. For clarity of depiction, the harmonic expansionlayers 133.1-133.5, 132.1-132.9, 131.1-131.6 and 110.1-110.5 are notshown to extend into the atmospheric material, although they areconsidered to do so. As can be seen in FIG. 1A, a structure having twomaterials occurring in a periodic direction will necessarily have twomaterials in an harmonic expansion layer.

[0020] With respect to semiconductors having a periodic structure with anominal two materials in periodic direction, it is often the case thatthe widths of the solid structures in the periodic direction isimportant to proper operation of the device being produced. For example,the width of a structure (such as a transistor gate) can determine howquickly or slowly a device will operate. Similarly, the width of aconductor can determine the resistance of the conductor, or the width ofa gap between two conductors can determine the amount of currentleakage. Furthermore, the geometry of a structure in the periodicdirection can also impact the geometry of successive layers of the chip.

[0021] Because the characteristic dimension of a structure in adirection orthogonal to the normal vector of the substrate generally hasthe most impact on the operation of a device and the fabrication of thecharacteristic dimension in successive layers of the chip, thatdimension is referred to as the “critical” dimension. Because of theimportance of critical dimension, it is common to use both the RCWAtechniques discussed above and various other types of microscopy (suchas critical-dimension scanning electron microscopy, cross-sectionalscanning electron microscopy, atomic force microscopy, and focused ionbeam measurement) to measure critical dimensions. While these techniquescan generally adequately measure critical dimensions of structureshaving a single solid material along a line in the periodic direction,none of these techniques can make accurate measurements of criticaldimensions of multiple material components of structures when more thana single solid material occurs in the periodic direction. In particular,such techniques generally cannot make accurate measurements of materialshaving more than two materials in a periodic direction.

[0022] However, a process which is intended to produce a structure withonly two materials per layer may have deviations which result in morethan two materials in a layer. For example, in FIG. 2A a semiconductordevice 810 is shown in which troughs 812 have been etched in a verticalportion 814, such as a series of ridges 815. Such a process nominallyproduces a structure having two materials along each line in theperiodic direction: the solid material of the ridges 815 and theatmospheric material in the troughs 812. However, as shown in FIG. 2B,which illustrates a common manufacturing defect on semiconductor device810, when etching the troughs 812, a thin polymer layer 818 can remaincoated on the side and bottom walls of the troughs 812.

[0023] Therefore, device 810 has three materials along the line 820 inthe periodic direction: the material of ridges 815, the material ofpolymer 818, and the atmospheric gas in trough 812. And, as noted above,techniques discussed above which can measure critical dimensions ofperiodic structures having a nominal two materials in the periodicdirection cannot be used to accurately measure the dimensions ofmultiple solid materials within structures having more than twomaterials in the periodic direction. Specifically, techniques ordinarilyused to measure the width of the ridges 815 will not yield an accuratemeasurement result when polymer 818 is present. This is because suchtechniques generally cannot distinguish between the material of ridges815 and the material of polymer 818.

[0024] A second example of a structure which is intended to nominallyhave only two materials in the periodic direction but which, due toadditional-material deviations, has more than two materials in theperiodic direction can occur in performing chemical mechanical polishing(“CMP”), as is shown in FIG. 3A. FIG. 3B shows a semiconductor device700 having a substrate 710 with a nitride layer 714 formed thereon.Troughs 712 are etched in the substrate 710 and nitride layer 714.Silicon dioxide plugs 716 are then placed in troughs 712. This resultsin a periodic structure which has either one or two materials in theperiodic direction. In particular, the substrate material 710 and thematerial of the silicon dioxide plugs 716 fall along line 722; thematerial of nitride layer 714 and the material of the silicon dioxideplugs 716 fall along line 718′; and the material of the substrate 710falls along line 724.

[0025] After the silicon dioxide plugs 716′ have been formed, such adevice 700 would typically be further processed using a techniquereferred to as “shallow trench isolation CMP”. This technique isintended to smooth the top face of the device so that the top of nitridelayer 714 and the top of the silicon dioxide plugs 716 both come to thesame level, shown by line 720. However, because silicon dioxide issofter than nitride, silicon dioxide plugs 716 will erode further thanthe nitride layer 714. This results in portions of silicon dioxide plugs716 dipping below the top surface of the nitride layer, and is known as“dishing” of silicon dioxide plugs 716. And, as shown in FIG. 3A alongline 718, near the top of the nitride layer 714, device 700 can hasthree materials occurring in the periodic direction: nitride, silicondioxide and the atmospheric material in those regions where the dishinghas resulted in the top surface 717 of the silicon dioxide plugs 716being below the level of line 718.

[0026] This type of deviation is referred to in the presentspecification as a “transverse” deviation because it is transverse tothe periodic direction of the structure and is transverse to what wouldgenerally be the direction along which the critical dimension ismeasured. That is, the deviation occurs in the direction normal to theface of device 700 (in a vertical direction in FIG. 3), rather thanalong the periodic direction. In contrast, the semiconductormanufacturing industry generally focuses on deviations in the criticaldimension, such as T-topping discussed earlier. Accordingly, the idea ofmeasuring the extent of any dishing occurring in a semiconductormanufacturing process has not generally arisen in the semiconductorfabrication industry since transverse deviations have not beenconsidered to have substantial effects on the operation of devices orthe fabrication of subsequent layers.

[0027] However, it is here predicted that with continuing technologicalinnovations allowing the size of semiconductor devices to steadilyshrink, the functioning of semiconductor devices will becomeincreasingly dependent on precise fabrication control and metrologyalong the transverse direction, and precise fabrication and control ofadditional-material deviations. Furthermore, recently developed deviceshave been designed with their critical dimension (i.e., the dimensionhaving the greatest effect on the operation of the device) along thenormal to the substrate, i.e., along the direction that the presentspecification has previously referred to as the transverse direction.Therefore, it is here predicted that future generations of semiconductorsystems will both have devices with their critical dimension parallel tothe substrate, and devices with their critical dimension perpendicularto the substrate.

SUMMARY OF THE INVENTION

[0028] A method and system in accordance with the present inventionallows measurement of semiconductor fabrication methods which ideallyhave only two materials along a line in a periodic direction, but whichhave deviations which result in more than two materials occurring alonga line in a periodic direction.

[0029] A method for metrology of additional-material structuraldeviations of a nominal periodic structure by comparison of a measureddiffraction spectrum from a target periodic structure with a calculateddiffraction spectrum from a hypothetical deviated periodic structure,where the hypothetical deviated periodic structure is defined byapplying the additional-material structural deviations to said nominalperiodic structure. The hypothetical deviated periodic structure has adirection of periodicity x, a direction of essentially-infiniteextension y which is orthogonal to the x direction, and a normaldirection z which is orthogonal to both the x and y directions. Aplurality of layers are defined parallel to an x-y plane. An x-z planecross-section of the periodic structure is sectioned into a plurality ofstacked rectangular sections such that only two materials from thenominal periodic structure are within each of the plurality of layersand at least three materials are within at least one of the plurality oflayers in the hypothetical deviated periodic structure. A harmonicexpansion of a function of the permittivity ∈ is performed along thedirection of periodicity x for each of the layers, including the layeror layers in the hypothetical deviated periodic structure whichinclude(s) at least three materials. Fourier space electromagneticequations are then set up in each of the layers using the harmonicexpansion of the function of the permittivity ∈ for each of the layersand Fourier components of electric and magnetic fields in each layer.The Fourier space electromagnetic equations are then coupled based onboundary conditions between the layers, and solved to provide thecalculated diffraction spectrum.

[0030] In a second aspect of the present invention, generation of thediffracted reflectivity of a periodic grating to determine values ofstructural properties of the periodic grating includes dividing theperiodic grating into a plurality of hypothetical layers at least one ofwhich is formed across at least first, second and third materials in theperiodic grating. Each hypothetical layer has its normal vectororthogonal to the direction of periodicity, and each hypothetical layerhas one of a plurality of possible combinations of hypothetical valuesof properties for that hypothetical layer. Sets of hypothetical layerdata are then generated. Each set of hypothetical layer data correspondsto a separate one of the plurality of hypothetical layers. The generatedsets of hypothetical layer data are processed to generate the diffractedreflectivity that would occur by reflecting electromagnetic radiationoff the periodic grating.

[0031] Preferably, each hypothetical layer is subdivided into aplurality of slab regions with each slab region corresponding to aseparate material within the hypothetical layer. Also, preferably,generating sets of hypothetical layer data includes expanding the realspace permittivity or the real space inverse permittivity of thehypothetical layers in a one-dimensional Fourier transformation alongthe direction of periodicity of the periodic grating. Preferably, theFourier transform is formulated as a sum over boundaries betweenmaterials in each layer.

[0032] In a third aspect of the present invention, a method ofgenerating an expression of the permittivity of a target periodicgrating having more than two materials in a periodic direction for usein an optical profilometry formalism for determining a diffractedreflectivity of the target periodic grating includes dividing the targetperiodic grating into a plurality of hypothetical layers. At least oneof the hypothetical layers is formed across each of at least a first,second and third material occurring along a line parallel to a directionof periodicity of the target periodic grating. At least one of theplurality of hypothetical layers is subdivided into a plurality ofhypothetical slabs to generate a plurality of hypothetical boundaries.Each of the plurality of hypothetical boundaries corresponds to anintersection of at least one of the plurality of hypothetical layerswith one of at least the first, second and third materials. Apermittivity function is determined for each of the plurality ofhypothetical layers. Then, a one-dimensional Fourier expansion of thepermittivity function of each hypothetical layer is completed along thedirection of periodicity of the target periodic grating by summing theFourier components over the plurality of hypothetical boundaries toprovide harmonic components of the at least one permittivity function. Apermittivity harmonics matrix is then defined including the harmoniccomponents of the Fourier expansion of the permittivity function.

[0033] A system of the present invention includes a microprocessorconfigured to perform the steps of the methods discussed above.Additionally, a computer readable storage medium in accordance with thepresent invention contains computer executable code for instructing acomputer to operate to complete the steps of the methods discussedabove.

BRIEF DESCRIPTION OF THE DRAWINGS

[0034]FIG. 1A is a cross-sectional view showing a periodic structurewhere there is a maximum of two materials along any line in the periodicdirection.

[0035]FIG. 1B is a close-up cross-sectional view of one of the ridges ofFIG. 1A with the ridge being sectioned into layers.

[0036]FIG. 2A is a cross-sectional view of a semiconductor device havingetched troughs and in which at most two materials occur along any linein the periodic direction.

[0037]FIG. 2B is a cross-sectional view of a the semiconductor deviceshown in FIG. 2A in which a residual polymer layer coats the etchedtroughs resulting in more than two materials unintentionally occurringalong any line in the periodic direction.

[0038]FIG. 3A is a cross-sectional view of a semiconductor device whichincludes transverse deviations resulting in more than two materialsoccurring a line in the periodic direction.

[0039]FIG. 3B is a cross-sectional view of the semiconductor device ofFIG. 3A without the transverse deviations which result in more than twomaterials occurring along a line in the periodic direction.

[0040]FIG. 4 shows a section of a diffraction grating labeled withvariables used in a mathematical analysis in accordance with the presentinvention.

[0041]FIG. 5 is a cross-sectional view of the semiconductor shown inFIG. 3A sectioned into harmonic expansion layers and discretized intorectangular slabs in accordance with the present invention.

[0042]FIG. 6 shows a process flow of a TE-polarization rigorouscoupled-wave analysis in accordance with the present invention.

[0043]FIG. 7 shows a process flow for a TM-polarization rigorouscoupled-wave analysis in accordance with the present invention.

[0044]FIG. 8 is a cross-sectional view of a drain region of asemiconductor device illustrating formation of spacers in a lightlydoped drain structure and having more than two materials along a line inthe periodic direction.

[0045]FIG. 9 is a flow chart illustrating a method of generating anexpression of the permittivity of a target periodic grating having morethan two materials in a periodic direction in accordance with thepresent invention.

[0046]FIG. 10 illustrates a computer system for implementation of thecomputation portions of the present invention.

DETAILED DESCRIPTION

[0047] The present invention relates to metrology of additional-materialdeviations and deviations in a direction transverse to the criticaldimension using a diffraction calculation technique. A system and methodin accordance with the present invention can be used for the measurementof one-dimensionally periodic surface profiles, particularly where thesurface profile has three or more materials along at least one line inthe periodic direction.

[0048]FIG. 4 is a diagram of a section of a periodic grating 600. Thesection of the grating 600 which is depicted includes three ridges 621which are shown as having a triangular cross-section. It should be notedthat the method of the present invention is applicable to cases wherethe ridges have shapes which are considerably more complex, and even tocases where the categories of “ridges” and “troughs” may be ill-defined.According to the lexography of the present specification, the term“ridge” will be used for one period of a periodic structure on asubstrate. Each ridge 621 of FIG. 4 is considered to extend infinitelyin the +y and −y directions, and an infinite, regularly-spaced series ofsuch ridges 621 are considered to extend in the +x and −x directions.The ridges 621 are atop a deposited film 610, and the film 610 is atop asubstrate 605 which is considered to extend semi-infinitely in the +zdirection. The normal vector {right arrow over (n)} to the grating is inthe −z direction.

[0049]FIG. 4 illustrates the variables associated with a mathematicalanalysis of a diffraction grating according to the present invention. Inparticular:

[0050] θ is the angle between the Poynting vector 630 of the incidentelectromagnetic radiation 631 and the normal vector n of the grating600. The Poynting vector 630 and the normal vector n define the plane ofincidence 640.

[0051] φ is the azimuthal angle of the incident electromagneticradiation 631, i.e., the angle between the direction of periodicity ofthe grating, which in FIG. 4 is along the x axis, and the plane ofincidence 640. (For ease of presentation, in the mathematical analysisof the present specification the azimuthal angle φ is set to zero.)

[0052] ψ is the angle between the electric-field vector {right arrowover (E)} of the incident electromagnetic radiation 631 and the plane ofincidence 640, i.e., between the electric field vector {right arrow over(E)} and its projection {right arrow over (E)}′on the plane of incidence640. When φ=0 and the incident electromagnetic radiation 631 ispolarized so that ψ=π/2, the electric-field vector {right arrow over(E)} is perpendicular to the plane of incidence 640 and themagnetic-field vector {right arrow over (H)} lies in the plane ofincidence 640, and this is referred to as the TE polarization. When φ=0and the incident electromagnetic radiation 631 is polarized so that ψ=0,the magnetic-field vector {right arrow over (H)} is perpendicular to theplane of incidence 640 and the electric-field vector {right arrow over(E)} lies in the plane of incidence 640, and this is referred to as theTM polarization. Any planar polarization is a combination of in-phase TEand TM polarizations. The method of the present invention describedbelow can be applied to any polarization which is a superposition of TEand TM polarizations by computing the diffraction of the TE and TMcomponents separately and summing them. Furthermore, although the‘off-axis’ φ≠0 case is more complex because it cannot be separated intoTE and TM components, the present invention is applicable to off-axisincident radiation as well.

[0053] λ is the wavelength of the incident electromagnetic radiation631.

[0054]FIG. 5 illustrates division of the periodic structure of FIG. 3Ainto a plurality of expansion layers to allow a mathematical analysis ofthe diffraction grating in accordance with the present invention. In thecoordinate system 211 shown in FIG. 5, the periodic direction is the xdirection, the transverse direction is the z direction, and the ydirection is a direction of essentially infinite extension orthogonal tothe x direction and z direction normal to the page.

[0055] As described above in reference to FIG. 3A, the periodicstructure 700 includes a substrate 710 with a nitride layer 714 formedthereon. Troughs 712 are etched in a periodic manner in the substrate710 and nitride layer 714. Silicon dioxide plugs 716 are then placed introughs 712. As explained in the Background section above, becausesilicon dioxide is softer than nitride, when a CMP process is applied tothe semiconductor device, silicon dioxide plugs 216 will erode furtherthan nitride layer 214. This results in portions of silicon dioxideplugs 216 dipping below the top surface of the nitride layer 714 andcreating a transverse deviation. In particular, near the top surface ofthe nitride layer 714, the semiconductor device has three materialsoccurring along a line in the periodic direction: nitride, silicondioxide and atmospheric gas.

[0056]FIG. 5 illustrates the variables associated with a mathematicaldescription of the dimensions of exemplary grating 700 according to thepresent invention. The nominal profile of FIG. 5 (i.e., the profile thatwould occur in this case if there was no dishing) has one or twomaterials per layer: the material of the substrate 710 and the silicondioxide of the plugs 716 in layers 225.4 and 225.5; the material of thesubstrate 710 in layers 225.6 and 225.7; the material of the substrate710 and the nitride of the nitride layer 714 in layers 225.1, 225.2 and225.3; and the material of the substrate 710 in layer 225.0. The dishingof the plugs 716 is considered to be an additional-material deviationprior to discretization. Also, atmospheric slabs 244.1 c and 244.2 c areconsidered to be additional-material deviations of the discretizedprofile. Accordingly, in layers 225.1 and 225.2 there are threematerials: the atmospheric material in slabs 244.1 c and 244.2 c, thenitride in slabs 238.1 and 238.2, and the silicon dioxide in slabs 244.1a and 244.1 b.

[0057]FIG. 9 is a flow chart illustrating a method of generating adiffracted reflectivity of a target periodic grating, such as grating700 of FIG. 5, having additional material deviations resulting in agrating with 2 or more materials occurring along a periodic direction.Specifically, FIG. 9 illustrates a method in accordance with the presentinvention for expressing the permittivity of a target periodic gratinghaving 2or more materials occurring along a periodic direction, thisexpression is referred to herein generally as hypothetical layer data.As expressed, this hypothetical layer data can be used to generate atheoretical or simulated diffracted reflectivity of the target periodicgrating. FIGS. 6 and 7 illustrate the details of a method in accordancewith the present invention of determining a theoretical diffractedreflectivity of a target periodic grating using the hypothetical layerdata. FIG. 6 illustrates this process flow for TE-polarization rigorouscoupled-wave analysis in accordance with the present invention and FIG.7 illustrates this process flow for a TM-polarization rigorouscoupled-wave analysis in accordance with the present invention.

[0058] Referring first to FIG. 9, in step 10, the target periodicgrating 700 (shown in FIG. 5) is divided into hypothetical harmonicexpansion layers. Referring again to FIG. 5, L+1 is the number of theharmonic expansion layers into which the system is divided. Harmonicexpansion layers 0 and L are considered to be semi-infinite layers.Harmonic expansion layer 0 is an “atmospheric” layer 701, such as vacuumor air, which typically has a refractive index n₀ near unity. Harmonicexpansion layer L is a “substrate” layer 710, which is typically siliconor germanium in semiconductor applications. In the case of the exemplarygrating 700, there are eight harmonic expansion layers, with theatmospheric layer 701 above grating 700 being the zeroth harmonicexpansion layer 225.0; the first and second harmonic expansion layers225.1 and 225.2, respectively, containing a top portion of the nitridelayer 714, the dished portion of the silicon dioxide plugs 712, and theatmospheric material; the third harmonic expansion layer 225.3containing the bottom portion of the nitride layer 714, and a middleportion of the silicon dioxide plugs 712; the third and fourth harmonicexpansion layers 225.3 and 225.4, respectively, containing the materialof the substrate 710 and the bottom portion of the silicon dioxide plugs716; and the sixth and seventh harmonic expansion layers 225.6 and 225.7containing only the material of the substrate 710. (Generically orcollectively, the harmonic expansion layers are assigned referencenumeral 225, and, depending on context, “harmonic expansion layers 225”may be considered to include the atmospheric layer 201 and/or thesubstrate 205.) As shown in FIG. 5, the harmonic expansion layers areformed parallel to the direction of periodicity of the grating 700. Itis also considered, however, that the layers form an angle with thedirection of periodicity of the grating being measured.

[0059] Referring again to FIG. 9, after dividing grating 700 into thehypothetical harmonic expansion layers as described above, in step 12,the hypothetical harmonic expansion layers are further divided intoslabs defined by the intersections of the harmonic expansion layers withthe materials forming the periodic grating. As shown in FIG. 5, thesection of each material within each intermediate harmonic expansionlayers 225.1 through 225.(L−1) is approximated by a planar slabs ofrectangular cross-section 238.1, 238.2, 238.3, 248.1, 248.2, 244.1 a,244.1 c, 244.1 b, 244.1 c, 244.2 c, 244.2 b, 244.2 c, 244.3, 244.4,244.5, 250.1, and 250.2. The top and bottom surfaces of each slab arelocated at the boundaries between harmonic expansion layers. The sidesurfaces of each slab are vertical and are located at the boundarybetween materials when that boundary is vertical, or across the boundarybetween materials when that boundary is not vertical. For instance, asshown in FIG. 5, slab 244.1 a has its left sidewall at the boundarybetween the nitride layer 714 and the plug 716. The right wall of slab244.1 a crosses the boundary between the plug 716 and the atmosphericmaterial at a point partway between leftmost edge of that boundary andrightmost edge of that boundary. Similarly, slab 244.1 b has its rightsidewall at the boundary between the nitride layer 714 and the plug 716,and the left wall of slab 244.1 b crosses the boundary between the plug716 and the atmospheric material at a point partway between leftmostedge of that boundary and rightmost edge of that boundary. Between slabs244.1 a and 244.1 b is slab 244.1 c. The left sidewall of slab 244.1 cis coincident with the right sidewall of slab 244.1 a, and the rightsidewall of slab 244.1 c is coincident with the left sidewall of slab244.1 b. Clearly, any geometry of exemplary grating 700 with across-section which does not consist solely of vertical and horizontalborders can be better approximated using a greater number of harmonicexpansion layers 225. For example, in practice, the portion of exemplarygrating 700 in harmonic expansion layers 225.1 and 225.2 might bedivided into a larger number of harmonic expansion layers so that thevertical sidewalls across the curved dishing surface 717 would be betterapproximated. However, for the sake of clarity, this region of exemplarygrating is divided into only two harmonic expansion layers 225.1 and225.2.

[0060] Other parameters shown in FIG. 5 are as follows:

[0061] D is the periodicity length or pitch, i.e., the length betweenequivalent points on pairs of adjacent ridges 221.

[0062] x^((l)) _(k) is the x coordinate of the starting border of thekth material in the lth layer, and x⁽¹⁾ _(k−1) is the x coordinate ofthe ending border of the kth material in the lth layer, so that X^((l))_(k)−x^((l)) _(k−1) is the width of the kth slab in the lth layer. Forexample, as shown in FIG. 5, x^((l)) ₂−x^((l)) ₁ is the width of thenitride slab 230.1.

[0063] t_(l) is the thickness of the lth layer 225.l for 1<l<(L−1). Thethicknesses t_(l) of the layers 225 are chosen so that (after thediscretization of the profile) every vertical line segment within eachlayer 225 passes through only a single material. For instance, prior todiscretization a vertical line in the region of slab 244.1 a would passthrough the boundary 717 between the atmospheric material and thesilicon dioxide. However, upon discretization, where that region isreplaced by a slab 244.1 a of silicon dioxide, any vertical line in thatregion only passes through the silicon dioxide

[0064] n_(k) is the index of refraction of the kth material in grating200.

[0065] In determining the diffraction generated by grating 200, asdiscussed in detail below, a Fourier space version of Maxwell'sequations is used. Referring again to FIG. 9, to generate theseequations, in step 14, hypothetical layer data is generated bycompleting a harmonic expansion of a function of the permittivities ofthe materials in the target periodic grating. In Step 14 a of FIG. 9(step 310 of FIG. 6 for TE polarization and step 410 of FIG. 7 for TMpolarization) the permittivities ε_(l)(x) for each layer l aredetermined or acquired as is known by those skilled in the art anddisclosed, for example, in U.S. patent application Ser. No. 09/728,146filed Nov. 28, 2000 entitled Profiler Business Model, by the presentinventors which is incorporated herein by reference in its entirety. Aone-dimensional Fourier transformation of the permittivity ε_(l)(x) orthe inverse permittivity π_(l)(x)=1/ε_(l)(x) of each layer l isperformed in step 14 b or FIG. 9 (step 312 of FIG. 6 and step 412 ofFIG. 7) along the direction of periodicity, x, of the periodic grating200 to provide the harmonic components of the permittivity ε_(l,i) orthe harmonic components of the inverse permittivity π_(1,i), where i isthe order of the harmonic component. In particular, the real-spacepermittivity ε_(l)(x) of the lth layer is related to the permittivityharmonics ε_(l,i) of the lth layer by $\begin{matrix}{{ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{ɛ_{l,i}{{\exp ( {j\frac{2\pi \quad i}{D}x} )}.}}}} & ( {1.1{.1}} )\end{matrix}$

[0066] Therefore, via the inverse transform, $\begin{matrix}{{ɛ_{0} = {\sum\limits_{k = 1}^{r}{n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}},} & ( {1.1{.2}} )\end{matrix}$

[0067] and for i not equal to zero, $\begin{matrix}{ɛ_{l,i} = {\sum\limits_{k = 1}^{r}{{\frac{n_{k}^{2}}{- {j2\pi}}\lbrack {( {{\cos ( {\frac{2\quad \pi}{D}x_{k}} )} - {\cos ( {\frac{2{\pi }}{D}x_{k - 1}} )}} ) - {j( {{\sin ( {\frac{2\quad \pi}{D}x_{k}} )} - {\sin ( {\frac{2\quad \pi}{D}x_{k - 1}} )}} )}} \rbrack}.}}} & ( {1.1{.3}} )\end{matrix}$

[0068] where the sum is over the number r of borders and n_(k) is theindex of refraction of the material between the kth and the (k−1)thborder and j is the imaginary number defined as the square root of −1.Similarly, the inverse of the permittivity, π_(l,i), of the lth layer isrelated to the inverse-permittivity harmonics π_(l,i), of the lth layerby $\begin{matrix}{{\pi_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{\pi_{l,i}{{\exp ( {j\frac{2\pi \quad i}{D}x} )}.}}}} & ( {1.1{.4}} )\end{matrix}$

[0069] Therefore, via the inverse transform, $\begin{matrix}{{\pi_{0} = {\sum\limits_{k = 1}^{r}{n_{k}^{- 2}\frac{x_{k - 1} - x_{k}}{D}}}},} & ( {1.1{.5}} )\end{matrix}$

[0070] and for i not equal to zero, $\begin{matrix}{\pi_{l,i} = {\sum\limits_{k = 1}^{r}{{\frac{n_{k}^{- 2}}{- {j2\pi}}\lbrack {( {{\cos ( {\frac{2\quad \pi}{D}x_{k}} )} - {\cos ( {\frac{2\pi \quad }{D}x_{k - 1}} )}} ) - {j( {{\sin ( {\frac{2\quad \pi}{D}x_{k}} )} - {\sin ( {\frac{2\quad \pi}{D}x_{k - 1}} )}} )}} \rbrack}.}}} & ( {1.1{.6}} )\end{matrix}$

[0071] where the sum is over the number r of borders and n_(k) is theindex of refraction of the material between the kth and the (k−1)thborder and j is the imaginary number defined as the square root of −1.It is important to note that equations for the harmonic components ofthe permittivity ε or inverse permittivity π provided by the prior artare formulated as a sum over materials, and are only directed towardsituations where each harmonic expansion layer has only one or twomaterials. In contrast, equations (1.1.2) and (1.1.3) and equations(1.1.5) and (1.1.6) are formulated as sums over the boundaries betweendifferent materials occurring in the periodic direction, and can handlegeometries with any number of materials in a harmonic expansion layer.

[0072] As such, the system and method of the present invention is notonly applicable to the semiconductor device 200 shown in FIG. 5, butalso to devices exhibiting other types of deviations, such assemiconductor device 810 shown in FIG. 2B which includes a polymerlayer.

[0073] Additionally, the system and method of the present inventioncould be used to measure structural dimensions of a periodic gratingwhich by design have three or more materials occurring along a line in aperiodic direction. One example of such a device is a field effecttransistor 740 shown in FIG. 8 having a source 742 a, a drain 742 b, anda gate 746. The gate 746 is placed on top of a insulating oxide barrierlayer 744 which coats the substrate 742. A voltage applied to the gate746 produces an electric field in the region between the source 742 aand the drain 742 b which strongly affects current flow between thesource 742 a and the drain 742 b. A top left spacer 748 at is formed onthe left side of the gate 746 on top of the barrier layer 744, a topright spacer 748 bt is formed on the right side of the gate 746 on topof the barrier layer 744, a bottom left spacer 748 ab is formed on theleft side of the gate 746 below the barrier layer 744 and to the rightof the source 742 a, and a bottom right spacer 748 ab is formed on theright side of the gate 746 below the barrier layer 744 and to the leftof the drain 742 b. The lower spacers 748 ab and 748 bb reduce themagnitude of electric field gradients in the regions near the source 742a and 742 b and below the barrier layer 744, thereby preventing theamount of current which ‘jumps’ through the barrier layer 744. The sizesand shapes of spacers 748 impacts the operation of device 740. If thespacers 748 at, 748 ab, 748 bt and 748 bb are too large, the operationof device 740 can be too slow. However, if the lower spacers 748 ab and748 bb are too small, current leakage through the barrier layer 744 canoccur, resulting in defective operation. Accordingly, it can beimportant to monitor the width of spacers 748. It should be noted thatthree materials lie along line 750: the material in the atmosphere 741,the material of the top spacers 748 at and 748 bt, and the material ofthe gate 746. Similarly, four materials lie along line 760: the materialof the substrate 742, the material of the lower spacers 748 ab and 748bb, the material of the source 742 a, and the material of the drain 742b.

[0074] The method disclosed herein of describing a periodic grating suchas a semiconductor device by dividing the grating into layers asdiscussed above and shown in FIG. 5, and expanding the permittivity of aperiodic grating such as grating 200 as shown in equations (1.1.1)through (1.1.3), or expanding the inverse permittivity as shown inequations (1.1.4) through (1.1.6), can be used with any opticalprofilometry formalism for determining a diffracted reflectivity thatuses a Fourier transform of the permittivity or inverse permittivity.

[0075] Referring again to FIG. 9, in step 16, the sets of hypotheticallayer data generated as described above are processed to generate thediffracted reflectivity. This step involves three general sub-steps:First, in sub-step 16 a, Fourier space electromagnetic field equationsare set up in each of the hypothetical layers using the harmonicexpansion of the permittivity function. Second, in sub-step 16 b, theseFourier space equations are coupled using boundary conditions betweenharmonic expansion layers. Finally, in sub-step 16 c, the coupledFourier space equations are solved to provide the diffractedreflectivity. Each of these sub-steps is explained in detail below withreference to the corresponding step in the flow charts of FIGS. 6 and 7.

[0076] To set up the Fourier space electromagnetic field equations, itis convenient to define the (2o+1)×(2o+1) Toeplitz-form, permittivityharmonics matrix E_(l) in step 14 c of FIG. 9. This permittivityharmonics matrix includes the harmonic components of the FourierExpansion of the permittivity ε_(l)(x) and is defined as:$\begin{matrix}{E_{l} = {\begin{bmatrix}ɛ_{l,0} & ɛ_{l,{- 1}} & ɛ_{l,{- 2}} & \ldots & ɛ_{l,{{- 2}o}} \\ɛ_{l,1} & ɛ_{l,0} & ɛ_{l,{- 1}} & \ldots & ɛ_{l,{- {({{2o} - 1})}}} \\ɛ_{l,2} & ɛ_{l,1} & ɛ_{l,0} & \ldots & ɛ_{l,{- {({{2o} - 2})}}} \\\ldots & \ldots & \ldots & \ldots & \ldots \\ɛ_{l,{2o}} & ɛ_{l,{({{2o} - 1})}} & ɛ_{l,{({{2o} - 2})}} & \ldots & ɛ_{l,0}\end{bmatrix}.}} & ( {1.1{.7}} )\end{matrix}$

[0077] A similar permittivity harmonics matrix is defined below inequation (2.1.4) which includes the harmonic components of the Fourierexpansion of the inverse permittivity π_(l)(x).

[0078] As will be seen below, to perform a TE-polarization calculationwhere oth-order harmonic components of the electric field {right arrowover (E)} and magnetic field {right arrow over (H)} are used, it isnecessary to use harmonics of the permittivity ε_(l,h) up to order 2o.

[0079] For the TE polarization, in the atmospheric layer the electricfield {right arrow over (E)} is formulated (324) as $\begin{matrix}{{\overset{->}{E}}_{0,y} = {\exp( {{{{- j}\quad k_{0}{n_{0}( {{\sin \quad \theta \quad x} + {\cos \quad \theta \quad z}} )}} + {\sum\limits_{i}{R_{i}{\exp ( {- {j( {{k_{xi}x} - {k_{0,{zi}}z}} )}} )}}}},} }} & ( {1.2{.1}} )\end{matrix}$

[0080] where the term on the left of the right-hand side of equation(1.2.1) is an incoming plane wave at an angle of incidence θ, the termon the right of the right-hand side of equation (1.2.1) is a sum ofreflected plane waves and R_(i) is the magnitude of the ith component ofthe reflected wave, and the wave vectors k₀ and (k_(xi), k_(0,zi)) aregiven by $\begin{matrix}{{k_{0} = {\frac{2\pi}{\lambda} = {\omega ( {\mu_{0}ɛ_{0}} )}^{1/2}}},} & ( {1.2{.2}} ) \\{{k_{xi} = {k_{0}( {{n_{0}{\sin (\theta)}} - {i( \frac{\lambda}{D} )}} )}},{and}} & ( {1.2{.3}} ) \\{k_{0,{zi}} = \{ {\begin{matrix}{k_{0}( {n_{l}^{2} - ( {k_{xi}/k_{0}} )^{2}} )}^{1/2} \\{{- j}\quad {k_{0}( {n_{l}^{2},( {k_{xi}/k_{0}} )^{2}} )}^{1/2}}\end{matrix}.} } & ( {1.2{.4}} )\end{matrix}$

[0081] where the value of k_(0,zi) is chosen from equation (1.2.4),i.e., from the top or the bottom of the expression, to provideRe(k_(0,zi))−Im(k_(0,zi))>0. This insures that k_(0,zi) ² i has apositive real part, so that energy is conserved. It is easily confirmedthat in the atmospheric layer 201, the reflected wave vector (k_(xi),k_(0,zi)) has a magnitude equal to that of the incoming wave vectork₀n₀. The magnetic field H in the atmospheric layer 201 is generatedfrom the electric field {right arrow over (E)} by Maxwell's equation(1.3.1) provided below.

[0082] The x-components k_(xi) of the outgoing wave vectors satisfy theFloquet condition (which is also called Bloch's Theorem, see Solid StatePhysics, N. W. Ashcroft and N. D. Mermin, Saunders College,Philadelphia, 1976, pages 133-134) in each of the layers (225 containingthe periodic ridges 221, and therefore, due to the boundary conditions,in the atmospheric layer 201 and the substrate layer 205 as well. Thatis, for a system having an n-dimensional periodicity given by$\begin{matrix}{{{f( \overset{arrow}{r} )} = {f( {\overset{arrow}{r} + {\sum\limits_{i = 1}^{n}{m_{i}{\overset{arrow}{d}}_{i}}}} )}},} & ( {1.2{.5}} )\end{matrix}$

[0083] where {right arrow over (d)}_(i) are the basis vectors of theperiodic system, and m_(i) takes on positive and negative integervalues, the Floquet condition requires that the wave vectors {rightarrow over (k)} satisfy $\begin{matrix}{{\overset{arrow}{k} = {{\overset{arrow}{k}}_{0} + {2\pi {\sum\limits_{i = 1}^{n}{m_{i}{\overset{arrow}{b}}_{i}}}}}},} & ( {1.2{.6}} )\end{matrix}$

[0084] where {right arrow over (b)}_(i) are the reciprocal latticevectors given by

({right arrow over (b)} _(i) ·{right arrow over (d)}_(j))=δ_(ij),  (1.2.7)

[0085] {right arrow over (k)}₀ is the wave vector of a free-spacesolution, and δ_(ij) is the Kronecker delta function. In the case of thelayers 225 of the periodic grating 200 of FIGS. 6A and 6B which have thesingle reciprocal lattice vector {right arrow over (b)} is {circumflexover (x)}/D, thereby providing the relationship of equation (1.2.3).

[0086] It may be noted that the formulation given above for the electricfield in the atmospheric layer 201, although it is an expansion in termsof plane waves, is not determined via a Fourier transform of areal-space formulation. Rather, the formulation is produced 324 a prioribased on the Floquet condition and the requirements that both theincoming and outgoing radiation have wave vectors of magnitude n₀k₀.Similarly, the plane wave expansion for the electric field in thesubstrate layer 205 is produced 324 a priori. In the substrate layer205, the electric field {right arrow over (E)} is formulated 324 as atransmitted wave which is a sum of plane waves where the x-componentsk_(xi) of the wave vectors (k_(xi), k_(0,z)) satisfy the Floquetcondition, i.e., $\begin{matrix}{{{\overset{arrow}{E}}_{L,y} = {\sum\limits_{i}{T_{i}{\exp ( {- {j( {{k_{xi}x} + {k_{L,{zi}}( {z - {\sum\limits_{l = 1}^{L - 1}t_{l}}} )}} )}} )}}}},{where}} & ( {1.2{.8}} ) \\{k_{L,{zi}} = \{ {\begin{matrix}{k_{0}( {n_{L}^{2} - ( {k_{xi}/k_{0}} )^{2}} )}^{1/2} \\{{- j}\quad {k_{0}( {n_{L}^{2} - ( {k_{xi}/k_{0}} )^{2}} )}^{1/2}}\end{matrix}.} } & ( {1.2{.9}} )\end{matrix}$

[0087] where the value of k_(L,zi) is chosen from equation (1.2.9),i.e., from the top or the bottom of the expression, to provideRe(k_(L,zi))−Im(k_(L,zi))>0, insuring that energy is conserved.

[0088] The plane wave expansions for the electric and magnetic fields inthe intermediate layers 225.1 through 225.(L−1) of FIG. 5 are also,referring again to FIG. 6, produced 334 a priori based on the Floquetcondition. The electric field {right arrow over (E)}_(l,y) in the lthlayer is formulated 334 as a plane wave expansion along the direction ofperiodicity, {circumflex over (x)}, i.e., $\begin{matrix}{{{\overset{arrow}{E}}_{l,y} = {\sum\limits_{i}{{S_{l,{yi}}(z)}{\exp ( {{- j}\quad k_{xi}x} )}}}},} & ( {1.2{.10}} )\end{matrix}$

[0089] where S_(l ,yi)(z) is the z-dependent electricfield harmonicamplitude for the lth layer and the ith harmonic. Similarly, themagnetic field {right arrow over (H)}_(l,y) in the lth layer isformulated 334 as a plane wave expansion along the direction ofperiodicity, {circumflex over (x)}, i.e., $\begin{matrix}{{{\overset{arrow}{H}}_{l,x} = {{- {j( \frac{ɛ_{0}}{\mu_{0}} )}^{1/2}}{\sum\limits_{i}{{U_{l,{xi}}(z)}{\exp ( {{- j}\quad k_{xi}x} )}}}}},} & ( {1.2{.11}} )\end{matrix}$

[0090] where U_(l,xi)(z) is the z-dependent magnetic field harmonicamplitude for the lth layer and the ith harmonic.

[0091] According to Maxwell's equations, the electric and magneticfields within a layer are related by $\begin{matrix}{{{\overset{arrow}{H}}_{l} = {( \frac{j}{\omega \quad \mu_{0}} ){\nabla{\times {\overset{arrow}{E}}_{l}}}}},{and}} & ( {1.3{.1}} ) \\{{\overset{arrow}{E}\quad}_{l} = {( \frac{- j}{\omega \quad ɛ_{0}{ɛ_{l}(x)}} ){\nabla{\times {{\overset{arrow}{H}}_{l}.}}}}} & ( {1.3{.2}} )\end{matrix}$

[0092] As discussed above with respect to FIG. 9, in sub-step 16 b theseFourier space equations are coupled using boundary conditions betweenthe harmonic expansion layers. Applying 342 the first Maxwell's equation(1.3.1) to equations (1.2.10) and (1.2.11) provides a first relationshipbetween the electric and magnetic field harmonic amplitudes S_(l) andU_(l) of the lth layer: $\begin{matrix}{\frac{\partial{S_{l,{yi}}(z)}}{\partial z} = {k_{0}{U_{l,{xi}}.}}} & ( {1.3{.3}} )\end{matrix}$

[0093] Similarly, applying 341 the second Maxwell's equation (1.3.2) toequations (1.2.10) and (1.2.11), and taking advantage of therelationship $\begin{matrix}{{k_{xi} + \frac{2\pi \quad h}{D}} = k_{x{({i - h})}}} & ( {1.3{.4}} )\end{matrix}$

[0094] which follows from equation (1.2.3), provides a secondrelationship between the electric and magnetic field harmonic amplitudesS_(l) and U_(l) for the lth layer: $\begin{matrix}{\frac{\partial U_{l,{xi}}}{\partial z} = {{( \frac{k_{xi}^{2}}{k_{0}} )S_{l,{yi}}} - {k_{0}{\sum\limits_{p}{ɛ_{({i - p})}{S_{l,{yp}}.}}}}}} & ( {1.3{.5}} )\end{matrix}$

[0095] While equation (1.3.3) only couples harmonic amplitudes of thesame order i, equation (1.3.5) couples harmonic amplitudes S_(l) andU_(l) between harmonic orders. In equation (1.3.5), permittivityharmonics ε_(i) from order −2o to +2o are required to couple harmonicamplitudes S_(l) and U_(l) of orders between −o and +o.

[0096] Combining equations (1.3.3) and (1.3.5) and truncating thecalculation to order o in the harmonic amplitude S provides 345 asecond-order differential matrix equation having the form of a waveequation, i.e., $\begin{matrix}{{\lbrack \frac{\partial^{2}S_{l,y}}{\partial z^{\prime 2}} \rbrack = {\lbrack A_{l} \rbrack \lbrack S_{l,y} \rbrack}},} & ( {1.3{.6}} )\end{matrix}$

[0097] z′=k₀ z, the wave-vector matrix [A_(l)] is defined as

[A _(l) ]=[K _(x)]² −[E _(l)],  (1.3.7)

[0098] where [K_(x)] is a diagonal matrix with the (i, i) element beingequal to (k_(xi)/k₀), the permittivity harmonics matrix [E_(l)] isdefined above in equation (1.1.4), and [S_(l,y)] and [∂S_(l,y)/∂z^(,2)]are column vectors with indices i running from −o to +o, i.e.,$\begin{matrix}{{\lbrack S_{l,y} \rbrack = \begin{bmatrix}S_{l,y,{({- o})}} \\\vdots \\S_{l,y,0} \\\vdots \\S_{l,y,o}\end{bmatrix}},} & ( {1.3{.8}} )\end{matrix}$

[0099] By writing 350 the homogeneous solution of equation (1.3.6) as anexpansion in pairs of exponentials, i.e., $\begin{matrix}{{{S_{l,{yi}}(z)} = {\sum\limits_{m = 1}^{{2o} + 1}{w_{l,i,m}\lbrack {{{c1}_{l,m}{\exp ( {{- k_{0}}q_{l,m}z} )}} + {{c2}_{l,m}{\exp ( {k_{0}{q_{l,m}( {z - t_{l}} )}} )}}} \rbrack}}},} & ( {1.3{.9}} )\end{matrix}$

[0100] its functional form is maintained upon second-orderdifferentiation by z′, thereby taking the form of an eigen equation.Solution 347 of the eigen equation

[A _(l) ][W _(l)]=[τ_(l) ][W _(l)]  (1.3.10)

[0101] provides 348 a diagonal eigenvalue matrix [τ_(l)] formed from theeigenvalues τ_(l,m) of the wave-vector matrix [A_(l)], and aneigenvector matrix [W_(l)] of entries w_(l,i,m), where w_(l,i,m) is theith entry of the mth eigenvector of [A_(l)]. A diagonal root-eigenvaluematrix [Q_(l)] is defined to be diagonal entries q_(l,i) which are thepositive real portion of the square roots of the eigenvalues τ_(l,i).The constants c1 and c2 are, as yet, undetermined.

[0102] By applying equation (1.3.3) to equation (1.3.9) it is found that$\begin{matrix}{{U_{l,{xi}}(z)} = {\sum\limits_{m = 1}^{{2o} + 1}{v_{l,i,m}\lbrack {{{- {c1}_{l,m}}{\exp ( {{- k_{0}}q_{l,m}z} )}} + {{c2}_{l,m}{\exp ( {k_{0}{q_{l,m}( {z - t_{l}} )}} )}}} \rbrack}}} & ( {1.3{.11}} )\end{matrix}$

[0103] (1.3.11)

[0104] where v_(l,i,m)=q_(l,m)w_(l,i,m). The matrix [V_(l)], to be usedbelow, is composed of entries v_(l,i,m).

[0105] The constants c1 and c2 in the homogeneous solutions of equations(1.3.9) and (1.3.11) are determined by applying 355 the requirement thatthe tangential electric and magnetic fields be continuous at theboundary between each pair of adjacent layers 225.1 and 225.(l+1). Atthe boundary between the atmospheric layer 201 and the first layer225.1, continuity of the electric field E_(y) and the magnetic fieldH_(x) requires $\begin{matrix}{{\begin{bmatrix}\delta_{i0} \\{j\quad n_{0}{\cos (\theta)}\delta_{i0}}\end{bmatrix} + {\begin{bmatrix}I \\{{- j}\quad Y_{0}}\end{bmatrix}R}} = {\begin{bmatrix}W_{1} & {W_{1}X_{1}} \\V_{1} & {{- V_{1}}X_{1}}\end{bmatrix}\begin{bmatrix}{c1}_{1} \\{c2}_{1}\end{bmatrix}}} & ( {1.4{.1}} )\end{matrix}$

[0106] where Y₀ is a diagonal matrix with entries (k_(0,zi)/k₀), X_(l)is a diagonal layer-translation matrix with elements exp(−k₀q_(l,m)t_(l)), R is a vector consisting of entries from R_(−o) toR_(+o), and c1 ₁ and c2 ₁ are vectors consisting of entries from c1_(1,0) and C1 _(1,2 o+1), and c2 _(1,0) and c2 _(1,2 o+1), respectively.The top half of matrix equation (1.4.1) provides matching of theelectric field E_(y) across the boundary of the atmospheric layer 225.0and the first layer 225.1, the bottom half of matrix equation (1.4.1)provides matching of the magnetic field H_(x) across the layer boundarybetween layer 225.0 and layer 125.1, the vector on the far left is thecontribution from the incoming radiation 631, shown in FIG. 4, in theatmospheric layer 201 of FIG. 5, the second vector on the left is thecontribution from the reflected radiation 132, shown in FIG. 4, in theatmospheric layer 201 of FIG. 5, and the portion on the right representsthe fields E_(y) and H_(x) in the first layer 225.1 of FIG. 5.

[0107] At the boundary between adjacent intermediate layers 225.l and225.(l+1), continuity of the electric field E_(y) and the magnetic fieldH_(x) requires $\begin{matrix}{{{\begin{bmatrix}{W_{l - 1}X_{l - 1}} & W_{l - 1} \\{W_{l - 1}X_{l - 1}} & {- V_{l - 1}}\end{bmatrix}\begin{bmatrix}{c1}_{l - 1} \\{c2}_{l - 1}\end{bmatrix}} = {\begin{bmatrix}W_{l} & {W_{l}X_{l}} \\V_{l} & {{- V_{l}}X_{l}}\end{bmatrix}\begin{bmatrix}{c1}_{l} \\{c2}_{l}\end{bmatrix}}},} & ( {1.4{.2}} )\end{matrix}$

[0108] where the top and bottom halves of the vector equation providematching of the electric field E_(y) and the magnetic field H_(x),respectively, across the l-1/l layer boundary.

[0109] At the boundary between the (L−1)th layer 225.(L−1) and thesubstrate layer 205, continuity of the electric field E_(y) and themagnetic field H_(x) requires $\begin{matrix}{{{\begin{bmatrix}{W_{L - 1}X_{L - 1}} & W_{L - 1} \\{W_{L - 1}X_{L - 1}} & {- V_{L - 1}}\end{bmatrix}\begin{bmatrix}{c1}_{L - 1} \\{c2}_{L - 1}\end{bmatrix}} = {\begin{bmatrix}I \\{jY}_{L}\end{bmatrix}T}},} & ( {1.4{.3}} )\end{matrix}$

[0110] where, as above, the top and bottom halves of the vector equationprovides matching of the electric field E_(y) and the magnetic fieldH_(x), respectively. In contrast with equation (1.4.1), there is only asingle term on the right since there is no incident radiation in thesubstrate 205.

[0111] Referring again to FIG. 6, matrix equation (1.4.1), matrixequation (1.4.3), and the (L−1) matrix equations (1.4.2) can be combined360 to provide a boundary-matched system matrix equation $\begin{matrix}{\begin{bmatrix}{- I} & W_{1} & {W_{1}X_{1}} & 0 & 0 & \ldots & \quad & \quad \\{j\quad Y_{0}} & V_{1} & {- {VX}} & 0 & 0 & \ldots & \quad & \quad \\0 & {{- W_{1}}X_{1}} & {- W_{1}} & W_{2} & {W_{2}X_{2}} & 0 & 0 & \ldots \\0 & {{- V_{1}}X_{1}} & V_{1} & V_{2} & {{- V_{2}}X_{2}} & 0 & 0 & \ldots \\0 & 0 & ⋰ & \quad & ⋰ & \quad & \quad & \vdots \\0 & {\quad 0} & {⋰\quad} & \quad & ⋰ & \quad & \quad & {\vdots \quad} \\\quad & \quad & {\quad \cdots} & \quad & {{- W_{L - 1}}X_{L - 1}} & {- W_{L - 1}} & {\quad I} & \quad \\\quad & \quad & \quad & \quad & {{- V_{L - 1}}X_{L - 1}} & V_{L - 1} & {\quad {j\quad Y_{L}}} & \quad\end{bmatrix}{\quad{\quad{{\begin{bmatrix}R \\{c1}_{1} \\{c2}_{1} \\\vdots \\\vdots \\{c1}_{L - 1} \\{c2}_{L - 1} \\T\end{bmatrix} = \begin{bmatrix}\delta_{i0} \\{j\quad \delta_{i0}n_{0}{\cos (\theta)}} \\0 \\\vdots \\\quad \\\quad \\\vdots \\0\end{bmatrix}},}}}} & ( {1.4{.4}} )\end{matrix}$

[0112] As is well understood by those skilled in the art, thisboundary-matched system matrix equation (1.4.4) may be solved 365(sub-step 16 c in the flow chart of FIG. 9) to provide the reflectivityR_(i) for each harmonic order i. (Alternatively, the partial-solutionapproach described in “Stable Implementation of the RigorousCoupled-Wave Analysis for Surface-Relief Dielectric Gratings: EnhancedTransmittance Matrix Approach”, E. B. Grann and D. A. Pommet, J. Opt.Soc. Am. A, vol. 12, 1077-1086, May 1995, can be applied to calculateeither the diffracted reflectivity R or the diffracted transmittance T.)

[0113] As noted above any planar polarization is a combination ofin-phase TE and TM polarizations. The method of the present inventioncan be applied to any polarization which is a superposition of TE and TMpolarizations by computing the diffraction of the TE and TM componentsseparately and summing them.

[0114] The method 400 of calculation for the diffracted reflectivity ofTM-polarized incident electromagnetic radiation shown in FIG. 7parallels that method 300 described above and shown in FIG. 6 for thediffracted reflectivity of TE-polarized incident electromagneticradiation. Referring to FIG. 4, for TM-polarized incident radiation 631the electric field vector {right arrow over (E)} is in the plane ofincidence 640, and the magnetic field vector {right arrow over (H)} isperpendicular to the plane of incidence 640. (The similarity in the TE-and TM-polarization RCWA calculations and the application of the presentinvention motivates the use of the term ‘electromagnetic field’ in thepresent specification to refer generically to either or both theelectric field and/or the magnetic field of the electromagneticradiation.)

[0115] As above, once the permittivity ε_(l)(x) is determined oracquired 410, the permittivity harmonics ε_(l,i) are determined 412using Fourier transforms according to equations (1.1.2) and (1.1.3), andthe permittivity harmonics matrix E_(l) is assembled as per equation(1.1.4). In the case of TM-polarized incident radiation 631, it has beenfound that the accuracy of the calculation may be improved byformulating the calculations using inverse-permittivity harmonicsπ_(l,i), since this will involve the inversion of matrices which areless singular. In particular, the one-dimensional Fourier expansion 412for the inverse of the permittivity ε_(l)(x) of the lth layer is givenby $\begin{matrix}{\frac{1}{ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{\pi_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}}}} & ( {2.1{.1}} )\end{matrix}$

[0116] Therefore, via the inverse Fourier transform this provides$\begin{matrix}{{\pi_{l,0} = {\sum\limits_{k = 1}^{r}{\frac{1}{n_{k}^{2}}\frac{x_{k} - x_{k - 1}}{D}}}},} & ( {2.1{.2}} )\end{matrix}$

[0117] and for i not equal to zero, $\begin{matrix}{\pi_{l,i} = {\sum\limits_{k = 1}^{r}{\frac{1}{{- {ji2}}\quad \pi}\frac{1}{n_{k}^{2}}( {( {{\cos ( {\frac{2\pi \quad i}{D}x_{k}} )} - {\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}} ) - {j( {{\sin ( {\frac{2\pi \quad i}{D}x_{k}} )} - {\sin ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}} )}} )}}} & ( {2.1{.3}} )\end{matrix}$

[0118] where the sum is over the number r of borders and n_(k) is theindex of refraction of the material between the kth and the (k−1)thborder and j is the imaginary number defined as the square root of −1.

[0119] As noted above with respect to equations (1.1.11) through(1.1.13), (1.1.2.1) and (1.1.3.1), by describing a periodic grating suchas a semiconductor device by dividing the grating into layers asdiscussed above and shown in FIG. 5, further subdividing the layers intoareas where the layers intersect with one of the materials, andexpanding the inverse permittivity of a periodic grating such as grating200 as shown in equations (2.1.1) through (2.1.3), as described below, amethod and system in accordance with the present invention can be usedto determine the diffracted reflectivity of a semiconductor devicehaving transverse or other defects, such as a polymer coating residueformed from an etching operation and illustrated in FIG. 2B, resultingin three or more materials per layer in a periodic direction.Additionally, the method disclosed herein of describing a periodicgrating such as a semiconductor device by dividing the grating intolayers as discussed above and shown in FIG. 5, and expanding the inversepermittivity of a periodic grating such as grating 200 as shown inequations (2.1.1) through (2.1.3) can be used with any opticalprofilometry formalism for determining a diffracted reflectivity thatuses a Fourier transform of the inverse permittivity.

[0120] The inverse-permittivity harmonics matrix P_(l) is defined as$\begin{matrix}{{P_{l} = \begin{bmatrix}\pi_{l,0} & \pi_{l,{- 1}} & \pi_{l,{- 2}} & \ldots & \pi_{l,{{- 2}o}} \\\pi_{l,1} & \pi_{l,0} & \pi_{l,{- 1}} & \ldots & \pi_{l,{- {({{2o} - 1})}}} \\\pi_{l,2} & \pi_{l,1} & \pi_{l,0} & \ldots & \pi_{l,{- {({{2o} - 2})}}} \\\ldots & \ldots & \ldots & \ldots & \ldots \\\pi_{l,{2o}} & \pi_{l,{({{2o} - 1})}} & \pi_{l,{({{2o} - 2})}} & \ldots & \pi_{l,0}\end{bmatrix}},} & \text{(2.1.4)}\end{matrix}$

[0121] where 2o is the maximum harmonic order of the inversepermittivity π_(l,i) used in the calculation. As with the case of the TEpolarization, for electromagnetic fields {right arrow over (E)} and{right arrow over (H)} calculated to order o it is necessary to useharmonic components of the permittivity ε_(l,i) and inverse permittivityπ_(l,i) to order 2o.

[0122] In the atmospheric layer the magnetic field {right arrow over(H)} is formulated 424 a priori as a plane wave incoming at an angle ofincidence θ, and a reflected wave which is a sum of plane waves havingwave vectors (k_(xi), k_(0,zi)) satisfying the Floquet condition,equation (1.2.6). In particular, $\begin{matrix}{{\overset{arrow}{H}}_{0,y} = {\exp( {{{{- j}\quad k_{0}{n_{0}( {{\sin \quad \theta \quad x} + {\cos \quad \theta \quad z}} )}} + {\sum\limits_{i}{R_{i}{\exp ( {- {j( {{k_{xi}x} - {k_{0,{zi}}z}} )}} )}}}},} }} & ( {2.2{.1}} )\end{matrix}$

[0123] where the term on the left of the right-hand side of the equationis the incoming plane wave, and R_(i) is the magnitude of the ithcomponent of the reflected wave. The wave vectors k₀ and (k_(xi),k_(o,zi)) are given by equations (1.2.2), (1.2.3), and (1.2.4) above,and, referring now to FIG. 5, the magnetic field {right arrow over (H)}in the atmospheric layer 201 is generated from the electric field {rightarrow over (E)} by Maxwell's equation (1.3.2). In the substrate layer205 the magnetic field {right arrow over (H)} is, as shown in FIG. 6,formulated 424 as a transmitted wave which is a sum of plane waves wherethe wave vectors (k_(xi), k_(0,zi)) satisfy the Floquet condition,equation (1.2.6), i.e., $\begin{matrix}{{{\overset{arrow}{H}}_{L,y} = {\sum\limits_{i}{T_{i}{\exp ( {- {j( {{k_{xi}x} + {k_{L,{zi}}( {z - {\sum\limits_{l = 1}^{L - 1}t_{l}}} )}} )}} )}}}},} & ( {2.2{.2}} )\end{matrix}$

[0124] where k_(L,zi) is defined in equation (1.2.9). Again based on theFloquet condition, the magnetic field {right arrow over (H)}_(l,y) inthe lth layer is formulated 434 as a plane wave expansion along thedirection of periodicity, {circumflex over (x)}, i.e., $\begin{matrix}{{{\overset{arrow}{H}}_{l,y} = {\sum\limits_{i}{{U_{l,{yi}}(z)}{\exp ( {{- j}\quad k_{xi}x} )}}}},} & ( {2.2{.3}} )\end{matrix}$

[0125] where U_(l,yi)(z) is the z-dependent magnetic field harmonicamplitude for the lth layer and the ith harmonic. Similarly, theelectric field {right arrow over (E)}_(l,x) in the lth layer isformulated 434 as a plane wave expansion along the direction ofperiodicity, i.e., $\begin{matrix}{{{\overset{arrow}{E}}_{l,x} = {{j( \frac{\mu_{0}}{ɛ_{0}} )}^{1/2}{\sum\limits_{i}{{S_{l,{xi}}(z)}{\exp ( {{- j}\quad k_{xi}x} )}}}}},} & ( {2.2{.4}} )\end{matrix}$

[0126] where S_(l,xi)(z) is the z-dependent electric field harmonicamplitude for the lth layer and the ith harmonic.

[0127] Substituting equations (2.2.3) and (2.2.4) into Maxwell'sequation (1.3.2) provides 441 a first relationship between the electricand magnetic field harmonic amplitudes S_(l) and U_(l) for the lthlayer: $\begin{matrix}{\frac{\partial\lbrack U_{l,{yi}} \rbrack}{\partial z^{\prime}} = {{\lbrack E_{l} \rbrack \lbrack S_{l,{xi}} \rbrack}.}} & ( {2.3{.1}} )\end{matrix}$

[0128] Similarly, substituting (2.2.3) and (2.2.4) into Maxwell'sequation (1.3.1) provides 442 a second relationship between the electricand magnetic field harmonic amplitudes S_(l) and U_(l) for the ithlayer: $\begin{matrix}{\frac{\partial\lbrack S_{l,{xi}} \rbrack}{\partial z^{\prime}} = {{( {{{\lbrack K_{x} \rbrack \lbrack P_{l} \rbrack}\lbrack K_{x} \rbrack} - \lbrack I\rbrack} )\lbrack U_{l,y} \rbrack}.}} & ( {2.3{.2}} )\end{matrix}$

[0129] where, as above, K_(x) is a diagonal matrix with the (i,i)element being equal to (k_(xi)/k₀). In contrast with equations (1.3.3)and (1.3.5) from the TE-polarization calculation, non-diagonal matricesin both equation (2.3.1) and equation (2.3.2) couple harmonic amplitudesS_(l) and U_(l) between harmonic orders.

[0130] Combining equations (2.3.1) and (2.3.2) provides a second-orderdifferential wave equation $\begin{matrix}{{\frac{\partial^{2}\lbrack U_{l,y} \rbrack}{\partial z^{\prime 2}} = {\{ {\lbrack E_{l} \rbrack ( {{{\lbrack K_{x} \rbrack \lbrack P_{l} \rbrack}\lbrack K_{x} \rbrack} - \lbrack I\rbrack} )} \} \lbrack U_{l,y} \rbrack}},} & ( {2.3{.3}} )\end{matrix}$

[0131] where [U_(l,y)] and [∂U_(l,y)/∂z^(,2)] are column vectors withindices running from −o to +o, and the permittivity harmonics [E_(l)] isdefined above in equation (1.1.7), and z′=k₀z. The wave-vector matrix[A_(l)] for equation (2.3.3) is defined as

[A _(l) ]=[E _(l)]([K _(x) ][P _(l) ][K _(x) ]−[I])  (2.3.4)

[0132] If an infinite number of harmonics could be used, then theinverse of the permittivity harmonics matrix [E_(l)] would be equal tothe inverse-permittivity harmonics matrix [P_(l)], and vice versa, i.e.,[E_(l)]⁻¹=[P_(l)], and [P_(l)]⁻¹=[E_(l)]. However, the equality does nothold when a finite number o of harmonics is used, and for finite o thesingularity of the matrices [E_(l)]⁻¹ and [P_(l)], and the singularityof the matrices [P_(l)]⁻¹ and [E_(l)], will generally differ. In fact,it has been found that the accuracy of RCWA calculations will varydepending on whether the wave-vector matrix [A_(l)] is defined as inequation (2.3.4), or

[A _(l) ]=[P _(l)]⁻¹([K _(x) ][E _(l)]⁻¹ [K _(x) ]−[I]),  (2.3.5)

[0133] or

[A _(l) ]=[E _(l)] ([K_(x) ][E _(l)]⁻¹[K_(x)]−[I]).  (2.3.6)

[0134] It should also be understood that although the case where

[A _(l) ]=[P _(l)]⁻¹([K _(x) ][P _(l) ] [K _(x) ]−[I])  (2.3.6′)

[0135] does not typically provide convergence which is as good as theformulations of equation (2.3.5) and (2.3.6), the present invention mayalso be applied to the formulation of equation (2.3.6°).

[0136] Regardless of which of the three formulations, equations (2.3.4),(2.3.5) or (2.3.6), for the wave-vector matrix [A_(l)] is used, thesolution of equation (2.3.3) is performed by writing 450 the homogeneoussolution for the magnetic field harmonic amplitude U_(l) as an expansionin pairs of exponentials, i.e., $\begin{matrix}{{U_{l,{yi}}(z)} = {\sum\limits_{m = 1}^{{2o} + 1}{{w_{l,i,m}\lbrack {{{c1}_{l,m}{\exp ( {{- k_{0}}q_{l,m}z} )}} + {{c2}_{l,m}{\exp ( {k_{0}{q_{l,m}( {z - t_{l}} )}} )}}} \rbrack}.}}} & ( {2.3{.7}} )\end{matrix}$

[0137] since its functional form is maintained upon second-orderdifferentiation by z′, and equation (2.3.3) becomes an eigen equation.Solution 447 of the eigen equation

[A _(l) ][W _(l)]=[τ_(l) ][W _(l)],  (2.3.8)

[0138] provides 448 an eigenvector matrix [W_(l)] formed from theeigenvectors w_(l,i) of the wave-vector matrix [A_(l)], and a diagonaleigenvalue matrix [τ_(l)] formed from the eigenvalues τ_(l,i) of thewave-vector matrix [A_(l)]. A diagonal root-eigenvalue matrix [Q_(l)] isformed of diagonal entries q_(l,i) which are the positive real portionof the square roots of the eigenvalues τ_(l,i). The constants c1 and c2of equation (2.3.7) are, as yet, undetermined.

[0139] By applying equation (1.3.3) to equation (2.3.5) it is found that$\begin{matrix}{{S_{l,{xi}}(z)} = {\sum\limits_{m = 1}^{{2o} + 1}{v_{l,i,m}\lbrack {{{- {c1}_{l,m}}{\exp ( {{- k_{0}}q_{l,m}z} )}} + {{c2}_{l,m}{\exp ( {k_{0}{q_{l,m}( {z - t_{l}} )}} )}}} \rbrack}}} & ( {2.3{.9}} )\end{matrix}$

[0140] (2.3.9)

[0141] where the vectors v_(l,i) form a matrix [V_(l)] defined as

[0142] [V]=[E]⁻¹[W][Q] when [A] is defined as in equation (2.3.4),(2.3.10)

[0143] [V]=[P][W][Q] when “ ” (2.3.5), (2.3.11)

[0144] [V]=[E]⁻¹[W][Q] when “ ” (2.3.6). (2.3.12)

[0145] The formulation of equations (2.3.5) and (2.3.11) typically hasimproved convergence performance (see P. Lalanne and G. M. Morris,“Highly Improved Convergence of the Coupled-Wave Method for TMPolarization”, J. Opt. Soc. Am. A, 779-784, 1996; and L. Li and C.Haggans, “Convergence of the coupled-wave method for metallic lamellardiffraction gratings”, J. Opt. Soc. Am. A, 1184-1189, June 1993)relative to the formulation of equations (2.3.4) and (2.3.11) (see M. G.Moharam and T. K. Gaylord, “Rigorous Coupled-Wave Analysis ofPlanar-Grating Diffraction”, J. Opt. Soc. Am., vol. 71, 811-818, July1981).

[0146] The constants c1 and c2 in the homogeneous solutions of equations(2.3.7) and (2.3.9) are determined by applying 455 the requirement thatthe tangential electric and tangential magnetic fields be continuous atthe boundary between each pair of adjacent layers (125.1)/(125.(l+1)),when the materials in each layer non-conductive. The calculation of thepresent specification is straightforwardly modified to circumstancesinvolving conductive materials, and the application of the method of thepresent invention to periodic gratings which include conductivematerials is considered to be within the scope of the present invention.Referring to FIG. 5, at the boundary between the atmospheric layer 201and the first layer 225.1, continuity of the magnetic field H_(y) andthe electric field E_(x) requires $\begin{matrix}{{\begin{bmatrix}\delta_{i0} \\{j\quad {\cos (\theta)}{\delta_{i0}/n_{0}}}\end{bmatrix} + {\begin{bmatrix}I \\{{- j}\quad Z_{0}}\end{bmatrix}R}} = {\begin{bmatrix}W_{1} & {W_{1}X_{1}} \\V_{1} & {{- V_{1}}X_{1}}\end{bmatrix}\begin{bmatrix}{c1}_{1} \\{c2}_{1}\end{bmatrix}}} & ( {2.4{.1}} )\end{matrix}$

[0147] where Z₀ is a diagonal matrix with entries (k_(0,zi)/n₀ ² k₀),X_(l) is a diagonal matrix with elements exp(−k_(0 q) _(l,m t) _(l)),the top half of the vector equation provides matching of the magneticfield Hy across the layer boundary, the bottom half of the vectorequation provides matching of the electric field E_(x) across the layerboundary, the vector on the far left is the contribution from incomingradiation in the atmospheric layer 201, the second vector on the left isthe contribution from reflected radiation in the atmospheric layer 201,and the portion on the right represents the fields H_(y) and E_(x) inthe first layer 225.1.

[0148] At the boundary between adjacent intermediate layers 225.l and225.(l+1), continuity of the electric field E_(y) and the magnetic fieldH_(x) requires $\begin{matrix}{{{\begin{bmatrix}{W_{l - 1}X_{l - 1}} & W_{l - 1} \\{W_{l - 1}X_{l - 1}} & {- V_{l - 1}}\end{bmatrix}\begin{bmatrix}{c1}_{l - 1} \\{c2}_{l - 1}\end{bmatrix}} = {\begin{bmatrix}W_{l} & {W_{l}X_{l}} \\V_{l} & {{- V_{l}}X_{l}}\end{bmatrix}\begin{bmatrix}{c1}_{l} \\{c2}_{l}\end{bmatrix}}},} & ( {2.4{.2}} )\end{matrix}$

[0149] where the top and bottom halves of the vector equation providesmatching of the magnetic field H_(y) and the electric field E_(x),respectively, across the layer boundary.

[0150] At the boundary between the (L−1)th layer 225.(L−1) and thesubstrate layer 205, continuity of the electric field E_(y) and themagnetic field H_(x) requires $\begin{matrix}{{{\begin{bmatrix}{W_{L - 1}X_{L - 1}} & W_{L - 1} \\{V_{L - 1}X_{L - 1}} & {- V_{L - 1}}\end{bmatrix}\begin{bmatrix}{c1}_{L - 1} \\{c2}_{L - 1}\end{bmatrix}} = {\begin{bmatrix}I \\{j\quad Z_{L}}\end{bmatrix}T}},} & ( {2.4{.3}} )\end{matrix}$

[0151] where, as above, the top and bottom halves of the vector equationprovides matching of the magnetic field H_(y) and the electric fieldE_(x), respectively. In contrast with equation (2.4.1), there is only asingle term on the right in equation (2.4.3) since there is no incidentradiation in the substrate 205.

[0152] Matrix equation (2.4.1), matrix equation (2.4.3), and the (L−1)matrix equations (2.4.2) can be combined 460 to provide aboundary-matched system matrix equation $\begin{matrix}{\begin{bmatrix}{- I} & W_{1} & {W_{1}X_{1}} & 0 & 0 & \ldots & \quad & \quad \\{j\quad Z_{0}} & V_{1} & {- {VX}} & 0 & 0 & \ldots & \quad & \quad \\0 & {{- W_{1}}X_{1}} & {- W_{1}} & W_{2} & {W_{2}X_{2}} & 0 & 0 & \ldots \\0 & {{- V_{1}}X_{1}} & V_{1} & V_{2} & {{- V_{2}}X_{2}} & 0 & 0 & \ldots \\0 & 0 & ⋰ & \quad & ⋰ & \quad & \quad & \vdots \\0 & {\quad 0} & {⋰\quad} & \quad & ⋰ & \quad & \quad & {\vdots \quad} \\\quad & \quad & {\quad \cdots} & \quad & {{- W_{L - 1}}X_{L - 1}} & {- W_{L - 1}} & {\quad I} & \quad \\\quad & \quad & \quad & \quad & {{- V_{L - 1}}X_{L - 1}} & V_{L - 1} & {\quad {j\quad Z_{L}}} & \quad\end{bmatrix}{\quad{\quad{{\begin{bmatrix}R \\{c1}_{1} \\{c2}_{1} \\\vdots \\\vdots \\{c1}_{L - 1} \\{c2}_{L - 1} \\T\end{bmatrix} = \begin{bmatrix}\delta_{i0} \\{j\quad \delta_{i0}{{\cos (\theta)}/n_{0}}} \\0 \\\vdots \\\quad \\\quad \\\vdots \\0\end{bmatrix}},}}}} & ( {2.4{.4}} )\end{matrix}$

[0153] As is well understood by those skilled in the art, theboundary-matched system matrix equation (2.4.4) may be solved 465 toprovide the reflectivity R for each harmonic order i. (Alternatively,the partial-solution approach described in “Stable Implementation of theRigorous Coupled-Wave Analysis for Surface-Relief Dielectric Gratings:Enhanced Transmittance Matrix Approach”, E. B. Grann and D. A. Pommet,J. Opt. Soc. Am. A, vol. 12, 1077-1086, May 1995, can be applied tocalculate either the diffracted reflectivity R or the diffractedtransmittance T.)

[0154] The matrix on the left in boundary-matched system matrixequations (1.4.4) and (2.4.4) is a square non-Hermetian complex matrixwhich is sparse (i.e., most of its entries are zero), and is of constantblock construction (i.e., it is an array of sub-matrices of uniformsize). The matrix can be stored in a database to provide computer accessfor solving for the diffracted reflectivity using numerical methods. Asis well known by those skilled in the art, the matrix can be storedusing the constant block compressed sparse row data structure (BSR)method (see S. Carney, M. Heroux, G. Li, R. Pozo, K. Remington and K.Wu, “A Revised Proposal for a Sparse BLAS Toolkit,”http://www.netlib.org, 1996). In particular, for a matrix composed of asquare array of square sub-matrices, the BSR method uses fivedescriptors:

[0155] B_LDA is the dimension of the array of sub-matrices;

[0156] O is the dimension of the sub-matrices;

[0157] VAL is a vector of the non-zero sub-matrices starting from theleftmost non-zero matrix in the top row (assuming that there is anon-zero matrix in the top row), and continuing on from left to right,and top to bottom, to the rightmost non-zero matrix in the bottom row(assuming that there is a non-zero matrix in the bottom row).

[0158] COL_IND is a vector of the column indices of the sub-matrices inthe VAL vector; and

[0159] ROW_PTR is a vector of pointers to those sub-matrices in VALwhich are the first non-zero sub-matrices in each row.

[0160] For example, for the left-hand matrix of equation (1.4.4), B_LDAhas a value of 2L, O has a value of 2o+1, the entries of VAL are (−I,W₁, W₁X₁, jY₀, V₁, −V₁X₁, −W₁X₁, −W₁, W₂, W₂X₂, −V₁X₁, V₁, V₂), theentries of COL_IND are (1, 2, 3, 1, 2, 3, 2, 3, 4, 5, 2, 3, 4, 5, . . .), and the entries of ROW_PTR are (1, 4, 7, 11, . . . ).

[0161] As is well-known in the art of the solution of matrix equations,the squareness and sparseness of the left-hand matrices of equations(1.4.4) and (2.4.4) are used to advantage by solving equations (1.4.4)and (2.4.4) using the Blocked Gaussian Elimination (BGE) algorithm. TheBGE algorithm is derived from the standard Gaussian Eliminationalgorithm (see, for example, Numerical Recipes, W. H. Press, B. P.Flannery, S. A. Teukolsky, and W. T. Vetterling, Cambridge UniversityPress, Cambridge, 1986, pp. 29-38) by the substitution of sub-matricesfor scalars. According to the Gaussian Elimination method, the left-handmatrix of equation (1.4.4) or (2.4.4) is decomposed into the product ofa lower triangular matrix [L], and an upper triangular matrix [U], toprovide an equation of the form

[L][U][x]=[b],  (3.1.1)

[0162] and then the two triangular systems [U][x]=[y] and [L][y]=[b] aresolved to obtain the solution [x]=[U]⁻¹[L]⁻¹[b], where, as per equations(1.4.4) and (2.4.4), [x ] includes the diffracted reflectivity R.

[0163] It should be noted that although the invention has been describedin term of a method, as per FIGS. 6, 7 and 9, the invention mayalternatively be viewed as an apparatus or system. Specifically, a shownin FIG. 10, the method of the present invention is preferablyimplemented on a computer system 900. Computer system 900 preferablyincludes information input/output (I/O) equipment 905, which isinterfaced to a computer 910. Computer 910 includes a central processingunit (CPU) 915, a cache memory 920 and a persistent memory 925, whichpreferably includes a hard disk, floppy disk or other computer readablemedium. The I/O equipment 905 typically includes a keyboard 902 and amouse 904 for the input of information, a display device 801 and aprinter 803. Many variations on computer system 900 are to be consideredwithin the scope of the present invention, including, withoutlimitation, systems with multiple I/O devices, multiple processors witha single computer, multiple computers connected by Internet linkages,and multiple computers connected by a local area network.

[0164] As is well understood by those skilled in the art, softwarecomputer code for implementing the steps of the method of the presentinvention illustrated in FIGS. 6, 7 and 9 and discussed in detail abovecan be stored in persistent memory 925. CPU 915 can then execute thesteps of the method of the present invention and store results ofexecuting the steps in cache memory 920 for completing diffractedreflectivity calculations as discussed above.

[0165] Referring to FIG. 4, it should also be understood that thepresent invention is applicable to off-axis or conical incidentradiation 631 (i.e., the case where φ≠0 and the plane of incidence 640is not aligned with the direction of periodicity, {circumflex over (x)},of the grating). The above exposition is straightforwardly adapted tothe off-axis case since, as can be seen in “Rigorous Coupled-WaveAnalysis of Planar-Grating Diffraction,” M. G. Moharam and T. K.Gaylord, J. Opt. Soc. Am., vol. 71, 811-818, July 1981, the differentialequations for the electromagnetic fields in each layer have homogeneoussolutions with coefficients and factors that are only dependent onintra-layer parameters and incident-radiation parameters.

[0166] It is also important to understand that, although the presentinvention has been described in terms of its application to the rigorouscoupled-wave method of calculating the diffraction of radiation, themethod of the present invention may be applied to any opticalprofilometry formalism where the system is divided into layers. Theforegoing descriptions of specific embodiments of the present inventionhave been presented for purposes of illustration and description. Theyare not intended to be exhaustive or to limit the invention to theprecise forms disclosed, and it should be understood that manymodifications and variations are possible in light of the aboveteaching. The embodiments were chosen and described in order to bestexplain the principles of the invention and its practical application,to thereby enable others skilled in the art to best utilize theinvention and various embodiments with various modifications as aresuited to the particular use contemplated. Many other variations arealso to be considered within the scope of the present invention.

[0167] Additionally, the calculation of the present specification isapplicable to circumstances involving conductive materials, ornon-conductive materials, or both, and the application of the method ofthe present invention to periodic gratings which include conductivematerials is considered to be within the scope of the present invention;the eigenvectors and eigenvalues of the matrix [A] may be calculatedusing another technique; the layer boundaries need not be planar andexpansions other than Fourier expansions, such as Bessel or Legendreexpansions, may be applied; the “ridges” and “troughs” of the periodicgrating may be ill-defined; the method of the present invention may beapplied to gratings having two-dimensional periodicity; the method ofthe present invention may be applied to any polarization which is asuperposition of TE and TM polarizations; the ridged structure of theperiodic grating may be mounted on one or more layers of films depositedon the substrate; the method of the present invention may be used fordiffractive analysis of lithographic masks or reticles; the method ofthe present invention may be applied to sound incident on a periodicgrating; the method of the present invention may be applied to medicalimaging techniques using incident sound or electromagnetic waves; themethod of the present invention may be applied to assist in real-timetracking of fabrication processes; the gratings may be made by ruling,blazing or etching; the method of the present invention may be utilizedin the field of optical analog computing, volume holographic gratings,holographic neural networks, holographic data storage, holographiclithography, Zernike's phase contrast method of observation of phasechanges, the Schlieren method of observation of phase changes, thecentral dark-background method of observation, spatial light modulators,acousto-optic cells, etc. In summary, it is intended that the scope ofthe present invention be defined by the claims appended hereto and theirequivalents.

What is claimed is:
 1. A method for metrology of additional-materialstructural deviations of a nominal periodic structure by comparison of ameasured diffraction spectrum from a target periodic structure with acalculated diffraction spectrum from a hypothetical deviated periodicstructure defined by applying said additional-material structuraldeviations to said nominal periodic structure, said hypotheticaldeviated periodic structure having a direction of periodicity x, adirection of essentially-infinite extension y orthogonal to saiddirection of periodicity x, and a normal direction z orthogonal to saiddirection of periodicity x and said direction of extension y,calculation of said calculated diffraction spectrum comprising the stepsof: defining a plurality of layers parallel to the x-y plane anddiscretizing an x-z plane cross-section of said periodic structure intoa plurality of stacked rectangular sections such that only two materialsfrom said nominal periodic structure are within each of said pluralityof layers, and only a single material lies along each line along thenormal direction z in each of said plurality of layers in saidhypothetical deviated periodic structure, and at least three materialsare within at least one of said plurality of layers in said hypotheticaldeviated periodic structure; performing a harmonic expansion of afunction of the permittivity ∈ along said direction of periodicity x foreach of said layers including said at least one of said plurality oflayers in said hypothetical deviated periodic structure having said atleast three materials therein; setting up Fourier space electromagneticequations in said each of said layers using said harmonic expansion ofsaid function of the permittivity ∈ for said each of said layers andFourier components of electric and magnetic fields; coupling saidFourier space electromagnetic equations based on boundary conditionsbetween said layers; and solving said coupling of said Fourier spaceelectromagnetic equations to provide said calculated diffractionspectrum.
 2. The method of claim 1 wherein said harmonic expansion ofsaid function of the permittivity ∈ along said direction of periodicityx for said at least one of said plurality of layers in said hypotheticaldeviated periodic structure having said at least three materials thereinis given by$ɛ_{0} = {\sum\limits_{k = 1}^{r}{n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}$

for the zeroth-order component, and$ɛ_{i} = {\sum\limits_{k = 1}^{r}{\frac{j\quad n_{k}^{2}}{\quad {{2}\quad \pi}}{\quad\lbrack {( {{\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )} - {\cos ( {\frac{2\pi \quad i}{D}x_{k}} )}} ) - {j( {{\sin ( {\frac{2\pi \quad i}{D}x_{k - 1}} )} - {\sin ( {\frac{2\pi \quad i}{D}x_{k}} )}} )}} \rbrack}}}$

for the i^(th)-order harmonic component, where D is the pitch of saidhypothetical deviated periodic structure, n_(k) is the index ofrefraction of a material between material boundaries at x_(k) andx_(k−l), j is the imaginary number defined as the square root of −1, andthere are r of said material boundaries within each period of saidhypothetical deviated periodic structure.
 3. The method of claim 1wherein said only two materials from said nominal periodic structurewithin at least a single one of said plurality of layers are a solid anda non-solid.
 4. The method of claim 1 wherein said periodic grating is asemiconductor grating with a critical dimension along said direction ofperiodicity x, and said additional-material structural deviations aredeviations along said normal direction z.
 5. The method of claim 4wherein said additional-material structural deviations along said normaldirection z result in an atmospheric region in what was a solid regionof said nominal periodic structure.
 6. The method of claim 1 whereinsaid periodic grating is a semiconductor grating with a criticaldimension along said direction of periodicity x, and saidadditional-material structural deviations are deviations along saiddirection of periodicity x due to a polymer deposit.
 7. The method ofclaim 1 wherein said periodic grating is a semiconductor grating andsaid additional-material structural deviations are purposefully includedto provide a structure having particular electronic characteristics. 8.The method of claim 1 wherein an initial one of said layers is anatmospheric region, and a final one of said layers is a substrate. 9.The method of claim 1 wherein said calculation of said calculateddiffraction spectrum is a rigorous coupled-wave calculation.
 10. Amethod of generating the diffracted reflectivity associated withdiffraction of electromagnetic radiation off a target periodic gratingto determine structural properties of the target periodic grating,including: dividing the target periodic grating into a plurality ofhypothetical layers, at least one of the hypothetical layers formedacross each of at least a first, second and third material, each of theat least first, second and third materials occurring along a directionof periodicity of the target periodic grating, each separatehypothetical layer having one of a plurality of possible combinations ofhypothetical values of properties for that hypothetical layer;generating sets of hypothetical layer data, each set of hypotheticallayer data corresponding to a separate one of the plurality ofhypothetical layers; and processing the generated sets of hypotheticallayer data to generate the diffracted reflectivity that would occur byreflecting electromagnetic radiation off the periodic grating.
 11. Themethod of claim 10 further including subdividing the hypothetical layersinto a plurality of slabs, each slab corresponding to the intersectionof one of the plurality of layers with one of at least the first, secondand third materials.
 12. The method of claim 11 wherein the step ofdividing the target periodic grating into a plurality of hypotheticallayers includes dividing the target periodic grating into a plurality ofhypothetical layers which are parallel to the direction of periodicityof the target periodic grating.
 13. The method of claim 10 wherein thestep of generating sets of hypothetical layer data includes expanding atleast one of either a function of a real space permittivity and afunction of a real space inverse permittivity of the hypothetical layersin a one-dimensional Fourier transformation along the direction ofperiodicity of the target periodic grating to provide harmoniccomponents of the at least one of either a function of a real spacepermittivity and a function of a real space inverse permittivity of thehypothetical layers.
 14. The method of claim 10 wherein the step ofgenerating sets of hypothetical layer data includes computing at leastone of: permittivity properties including a function of a permittivity∈₁(x) of each of the hypothetical layers of the target periodic grating,the harmonic components so ∈_(1,i) of the function of the permittivity∈₁(x), and a permittivity harmonics matrix [E_(l)]; andinverse-permittivity properties including a function of aninverse-permittivity π₁(x) of each of the hypothetical layers of thetarget periodic grating, the harmonic components π_(1,i) of the functionof the inverse-permittivity π₁(x), and an inverse-permittivity harmonicsmatrix [P_(l)].
 15. The method of claim 14 wherein the step ofprocessing the generated sets of hypothetical layer data includes:computing a wave-vector matrix [A_(l)] by combining a series expansionof the electric field of each of the hypothetical layers of the targetperiodic grating with at least one of at least the permittivityharmonics matrix [E_(l)] and inverse-permittivity harmonics matrix[P_(l)]; and computing the ith entry w_(l,i,m) of the mth eigenvector ofthe wave-vector matrix [A_(l)] and the mth eigenvalue τ_(l,m) of thewave-vector matrix [A_(l)] to form an eigenvector matrix [W_(l)] and aroot-eigenvalue matrix [Q_(l)].
 16. The method of claim 10 wherein thestep of generating sets of hypothetical layer data includes expandingone of at least a function of a permittivity ∈₁(x) and a function of aninverse-permittivity π_(l)(x)=1/∈₁(x) of the at least one of thehypothetical layers formed across each of at least the first, second andthird materials of the target periodic grating in a one-dimensionalFourier transformation, the expansion performed along the direction ofperiodicity of the target periodic grating according to at least one of:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{ɛ_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}}}$where  $ɛ_{l,i} = {\sum\limits_{k = 1}^{r}{\frac{n_{k}^{2}}{{- {ji2}}\quad \pi}{\quad\lbrack ( {{{\cos ( {\frac{2i\quad \pi}{D}x_{k}} )} - { { {\quad{\quad{\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}}} ) - {j( {{\sin ( {\frac{2i\quad \pi}{D}x_{k}} )} - {\sin ( {\frac{2i\quad \pi}{D}x_{k - 1}} )}} )}} \rbrack {and}\quad {\pi_{l}(x)}}} = {\frac{1}{ɛ_{l}(x)} = {{\sum\limits_{i = {- \infty}}^{\infty}{\pi_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}{where}\quad \text{}\pi_{l,i}}} = {\sum\limits_{k = 1}^{r}{\frac{1}{{- {ji}}\quad 2\quad \pi}\frac{1}{n_{k}^{2}}( {( {{\cos ( {\frac{2\pi \quad i}{D}x_{k}} )} - {\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}} ) - {j( {{\sin ( {\frac{2\pi \quad i}{D}x_{k}} )} - {\sin ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}} )}} )}}}}}  }}}$

where D is the pitch of said hypothetical deviated periodic structure,n_(k) is the index of refraction of a material between materialboundaries at x_(k) and X_(k−1),j is the imaginary number defined as thesquare root of −1, and there are r of said material boundaries withineach period of said hypothetical deviated periodic structure.
 17. Themethod of claim 10, wherein the step of processing the generated sets ofhypothetical layer data includes: constructing a matrix equation fromthe intermediate data corresponding to the hypothetical layers of thetarget periodic grating; and solving the constructed matrix equation todetermine the diffracted reflectivity value R_(i) for each harmonicorder i.
 18. A method of generating the diffracted reflectivityassociated with diffraction of electromagnetic radiation off a targetperiodic grating to determine structural properties of the targetperiodic grating, including: dividing the target periodic grating into aplurality of hypothetical layers, at least one of the hypotheticallayers formed across each of at least a first, second and third materialoccurring along a line parallel to a direction of periodicity of thetarget periodic grating; performing an harmonic expansion of a functionof the permittivity ∈ along the direction of periodicity of the targetperiod grating for each of the hypothetical layers including the atleast one of the plurality of layers formed across each of at least afirst, second and third material; setting up Fourier spaceelectromagnetic equations in each of the hypothetical layers using theharmonic expansion of the function of the permittivity ∈ for said eachof the hypothetical layers and Fourier components of electric andmagnetic fields; coupling the Fourier space electromagnetic equationsbased on boundary conditions between the layers; and solving thecoupling of the Fourier space electromagnetic equations to provide adiffracted reflectivity.
 19. The method of claim 18 further includingsubdividing at least one of the plurality of hypothetical layers into aplurality of hypothetical slabs, each hypothetical slab corresponding toan intersection of the at least one of the plurality of hypotheticallayers with one of at least the first, second and third materials. 20.The method of claim 19 wherein the step of subdividing at least one ofthe hypothetical layers into a plurality of hypothetical slabs includessubdividing the at least one hypothetical layer into a plurality ofhypothetical slabs such that only a single material lies along any lineperpendicular to the direction of periodicity of the target periodicgrating and normal to the target periodic grating.
 21. The method ofclaim 18 wherein the harmonic expansion of the function of thepermittivity along the direction of periodicity of the target periodicgrating for the at least one of the hypothetical layers formed acrosseach of at least the first, second and third material is given by:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{ɛ_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}}}$${where},\quad {ɛ_{0} = {\sum\limits_{k = 1}^{r}{n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}}$

is the zeroth-order component, and$ɛ_{i} = {\sum\limits_{k = 1}^{r}{\frac{j\quad n_{k}^{2}}{\quad {{i2}\quad \pi}}{\quad\lbrack  { ( {{\cos ( {\frac{2\quad \pi \quad i}{D}x_{k - 1}} )} - {\quad{\quad{\cos ( {\frac{2\pi \quad i}{D}x_{k}} )}}}}  ) - {j( {{\sin ( {\frac{2\pi \quad i}{D}x_{k - 1}} )} - {\sin ( {\frac{2\quad \pi \quad i}{D}x_{k}} )}} )}} \rbrack }}}$

is the i^(th)-order harmonic component, l indicates the lth one of theplurality of hypothetical layers, D is the pitch of said hypotheticaldeviated periodic structure, n_(k) is the index of refraction of amaterial between material boundaries at x_(k) and X_(k−1),j is theimaginary number defined as the square root of −1, and there are r ofsaid material boundaries within each period of said hypotheticaldeviated periodic structure.
 22. A method of generating an expression ofthe permittivity of a target periodic grating having more than twomaterials in a periodic direction for use in an optical profilometryformalism for determining a diffracted reflectivity of the targetperiodic grating comprising: dividing the target periodic grating into aplurality of hypothetical layers, at least one of the hypotheticallayers formed across each of at least a first, second and third materialoccurring along a line parallel to a direction of periodicity of thetarget periodic grating; subdividing at least one of the plurality ofhypothetical layers into a plurality of hypothetical slabs to generate aplurality of hypothetical boundaries, each of the plurality ofhypothetical boundaries corresponding to an intersection of at least oneof the plurality of hypothetical layers with one of at least the first,second and third materials; determining a permittivity function for eachof the plurality of hypothetical layers; completing a one-dimensionalFourier expansion of the permittivity function of each hypotheticallayer along the direction of periodicity of the target periodic gratingby summing the Fourier components over the plurality of hypotheticalboundaries to provide harmonic components of the at least onepermittivity function; and defining a permittivity harmonics matrixincluding the harmonic components of the Fourier expansion of thepermittivity function.
 23. The method of claim 22 wherein the step ofcompleting a one dimensional Fourier transform of a permittivityfunction includes completing the one dimensional Fourier transform ofthe permittivity function ∈₁(x) according to:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{ɛ_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}}}$where  $ɛ_{0} = {\sum\limits_{k = 1}^{r}{n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}$

is the zeroth order component,$ɛ_{l,i} = {\sum\limits_{k = 1}^{r}{\frac{n_{k}^{2}}{{- {ji2}}\quad \pi}{\quad\lbrack  { ( {{\cos ( {\frac{2\quad i\quad \pi}{D}x_{k}} )} - {\quad{\quad{\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}}}}  ) - {j( {{\sin ( {\frac{2\quad i\quad \pi}{D}x_{k}} )} - {\sin ( {\frac{2\quad i\quad \pi}{D}x_{k - 1}} )}} )}} \rbrack }}}$

is the ith order component, l indicates the ith one of the plurality ofhypothetical layers, D is the pitch of the target periodic grating,n_(k) is the index of refraction of a material between each of theplurality of hypothetical boundaries at x_(k) and x_(k−1), j is theimaginary number defined as the square root of −1, and there are rhypothetical boundaries within each period of the target periodicgrating.
 24. The method of claim 23 wherein the permittivity harmonicsmatrix E_(l) is a (2o+1)×(2o+1) Toeplitz-form matrix having the form:$E_{l} = \begin{bmatrix}ɛ_{l,0} & ɛ_{l,{- 1}} & ɛ_{l,{- 2}} & \ldots & ɛ_{l,{{- 2}o}} \\ɛ_{l,1} & ɛ_{l,0} & ɛ_{l,{- 1}} & \ldots & ɛ_{l,{- {({{2o} - 1})}}} \\ɛ_{l,2} & ɛ_{l,1} & ɛ_{l,0} & \ldots & ɛ_{l,{- {({{2o} - 2})}}} \\\ldots & \ldots & \ldots & \ldots & \ldots \\ɛ_{l,{2o}} & ɛ_{l,{({{2o} - 1})}} & ɛ_{l,{({{2o} - 2})}} & \ldots & ɛ_{l,0}\end{bmatrix}$

where o is the order of the harmonic component.
 25. The method of claim22 wherein the step of completing a one dimensional Fourier transform ofa permittivity function includes completing the one dimensional Fouriertransform of the permittivity function π_(l)(x) according to:${\pi_{l}(x)} = {\frac{1}{ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{\pi_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}}}}$where  $\pi_{l,0} = {\sum\limits_{k = 1}^{r}{\frac{1}{n_{k}^{2}}\frac{x_{k} - x_{k - 1}}{D}}}$

is the zeroth order component,$\pi_{l,i} = {\sum\limits_{k = 1}^{r}{\frac{1}{\quad {{- {ji2}}\quad \pi}}\frac{1}{n_{k}^{2}}{\quad(  { ( {{\cos ( {\frac{2\quad \pi \quad i}{D}x_{k}} )} - {\quad{\quad{\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}}}}  ) - {j( {{\sin ( {\frac{2\pi \quad i}{D}x_{k}} )} - {\sin ( {\frac{2\quad \pi \quad i}{D}x_{k - 1}} )}} )}} ) }}}$

is the ith order component, where l indicates the lth one of theplurality of hypothetical layers, D is the pitch of the target periodicgrating, n_(k) is the index of refraction of a material between each ofthe plurality of hypothetical boundaries at x_(k) and x_(k−1),j is theimaginary number defined as the square root of −1, and there are rhypothetical boundaries within each period of the target periodicgrating.
 26. The method of claim 25 wherein the permittivity harmonicsmatrix P_(l) is a (2o+1)×(2o+1) Toeplitz-form matrix having the form:$P_{l} = \begin{bmatrix}\pi_{l,0} & \pi_{l,{- 1}} & \pi_{l,{- 2}} & \ldots & \pi_{l,{{- 2}o}} \\\pi_{l,1} & \pi_{l,0} & \pi_{l,{- 1}} & \ldots & \pi_{l,{- {({{2o} - 1})}}} \\\pi_{l,2} & \pi_{l,1} & \pi_{l,0} & \ldots & \pi_{l,{- {({{2o} - 2})}}} \\\ldots & \ldots & \ldots & \ldots & \ldots \\\pi_{l,{2o}} & \pi_{l,{({{2o} - 1})}} & \pi_{l,{({{2o} - 2})}} & \ldots & \pi_{l,0}\end{bmatrix}$

where o is the order of the harmonic component.
 27. A computer readablestorage medium containing computer executable code for generating thetheoretical diffracted reflectivity associated with diffraction ofelectromagnetic radiation off a target periodic grating to determinestructural properties of the target periodic grating by instructing acomputer to operate as follows: divide the target periodic grating intoa plurality of hypothetical layers, at least one of the hypotheticallayers formed across each of at least a first, second and thirdmaterial, each of the at least first, second and third materialsoccurring along a direction of periodicity of the target periodicgrating, each separate hypothetical layer having one of a plurality ofpossible combinations of hypothetical values of properties for thathypothetical layer; generate sets of hypothetical layer data, each setof hypothetical layer data corresponding to a separate one of theplurality of hypothetical layers; and process the generated sets ofhypothetical layer data to generate the diffracted reflectivity thatwould occur by reflecting electromagnetic radiation off the periodicgrating.
 28. The computer readable storage medium of claim 27 whereinthe computer is further instructed to subdivide the hypothetical layersinto a plurality of slabs, each slab corresponding to the intersectionof one of the plurality of layers with one of at least the first, secondand third materials.
 29. The computer readable storage medium of claim28 wherein in dividing the target periodic grating into a plurality ofhypothetical layers the computer is instructed to divide the targetperiodic grating into a plurality of hypothetical layers which areparallel to the direction of periodicity of the target periodic grating.30. The computer readable medium of claim 27 wherein in generating setsof hypothetical layer data the computer is instructed to expand at leastone of either a function of a real space permittivity and a function ofa real space inverse permittivity of the hypothetical layers in aone-dimensional Fourier transformation along the direction ofperiodicity of the target periodic grating to provide harmoniccomponents of the at least one of either a function of a real spacepermittivity and a function of a real space inverse permittivity of thehypothetical layers.
 31. The computer readable storage medium of claim27 wherein in generating sets of hypothetical layer data the computer isinstructed to compute at least one of: permittivity properties includinga function of a permittivity ∈₁(x) of each of the hypothetical layers ofthe target periodic grating, the harmonic components ∈_(1,i) of thefunction of the permittivity ∈₁(x), and a permittivity harmonics matrix[E_(l)]; and inverse-permittivity properties including a function of aninverse-permittivity π₁(x) of each of the hypothetical layers of thetarget periodic grating, the harmonic components π_(1,i) of the functionof the inverse-permittivity π₁(x), and an inverse-permittivity harmonicsmatrix [P_(l)].
 32. The computer readable medium of claim 31 wherein inprocessing the generated sets of hypothetical layer data the computer isinstructed to: compute a wave-vector matrix [A_(l)] by combining aseries expansion of the electric field of each of the hypotheticallayers of the target periodic grating with at least one of at least thepermittivity harmonics matrix [E_(l)] and inverse-permittivity harmonicsmatrix [P_(l)]; and compute the ith entry w_(l,i,M) of the mtheigenvector of the wave-vector matrix [A_(l)] and the mth eigenvalueτ_(l,m) of the wave-vector matrix [A_(l)] to form an eigenvector matrix[W_(l)] and a root-eigenvalue matrix [Q_(l)].
 33. The computer readablemedium of claim 27 wherein in generating sets of hypothetical layer datathe computer is instructed to expand one of at least a permittivity∈₁(x) and an inverse-permittivity π_(l)(x)=1/∈₁(x) of the at least oneof the hypothetical layers formed across each of at least the first,second and third materials of the target periodic grating in aone-dimensional Fourier transformation, the expansion performed alongthe direction of periodicity of the target periodic grating according toat least one of:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{ɛ_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}}}$where    $ɛ_{l,i} = {\sum\limits_{k = 1}^{r}{\frac{n_{k}^{2}}{{- {ji2}}\quad \pi}{\quad\lbrack ( {{{\cos ( {\frac{2i\quad \pi}{D}x_{k}} )} - { { {\quad{\quad{\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}}} ) - {j( {{\sin ( {\frac{2i\quad \pi}{D}x_{k}} )} - {\sin ( {\frac{2i\quad \pi}{D}x_{k - 1}} )}} )}} \rbrack {and}\quad {\pi_{l}(x)}}} = {\frac{1}{ɛ_{l}(x)} = {{\sum\limits_{i = {- \infty}}^{\infty}{\pi_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}{where}\quad \text{}\pi_{l,i}}} = {\sum\limits_{k = 1}^{r}{\frac{1}{{- {ji2}}\quad \pi}\frac{1}{n_{k}^{2}}( {( {{\cos ( {\frac{2\pi \quad i}{D}x_{k}} )} - {\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}} ) - {j( {{\sin ( {\frac{2\pi \quad i}{D}x_{k}} )} - {\sin ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}} )}} )}}}}}  }}}$

where l indicates the lth one of the plurality of hypothetical layers, Dis the pitch of said hypothetical deviated periodic structure, n_(k) isthe index of refraction of a material between material boundaries atx_(k) and X_(k−1),j is the imaginary number defined as the square rootof −1 and there are r of said material boundaries within each period ofsaid hypothetical deviated periodic structure.
 34. The computer readablemedium of claim 27, wherein in processing the generated sets ofhypothetical layer data the computer is configured to: construct amatrix equation from the intermediate data corresponding to thehypothetical layers of the target periodic grating; and solve theconstructed matrix equation to determine the diffracted reflectivityvalue R_(i) for each harmonic order i.
 35. A computer readable storagemedium containing computer executable code for generating the diffractedreflectivity associated with diffraction of electromagnetic radiationoff a target periodic grating to determine structural properties of thetarget periodic grating by instructing a computer to operate as follows:divide the target periodic grating into a plurality of hypotheticallayers, at least one of the hypothetical layers formed across each of atleast a first, second and third material occurring along a line parallelto a direction of periodicity of the target periodic grating; perform anharmonic expansion of a function of the permittivity ∈₁(x) along thedirection of periodicity of the target period grating for each of thehypothetical layers including the at least one of the plurality oflayers formed across each of at least a first, second and thirdmaterial; set up Fourier space electromagnetic equations in each of thehypothetical layers using the harmonic expansion of the function of thepermittivity ∈₁(x) for said each of the hypothetical layers and Fouriercomponents of electric and magnetic fields; couple the Fourier spaceelectromagnetic equations based on boundary conditions between thelayers; and solve the coupling of the Fourier space electromagneticequations to provide a diffracted reflectivity.
 36. The computerreadable storage medium of claim 35 wherein the computer is furtherinstructed to subdivide at least one of the plurality of hypotheticallayers into a plurality of hypothetical slabs, each hypothetical slabcorresponding to an intersection of the at least one of the plurality ofhypothetical layers with one of at least the first, second and thirdmaterials.
 37. The computer readable storage medium of claim 36 whereinin subdividing the at least one of the hypothetical layers into aplurality of hypothetical slabs the computer executable code instructsthe computer to subdivide the at least one hypothetical layer into aplurality of hypothetical slabs such that only a single material liesalong any line perpendicular to the direction of periodicity of thetarget periodic grating and normal to the target periodic grating. 38.The computer readable storage medium of claim 37 wherein in performingan harmonic expansion of a function of the permittivity ∈₁(x) thecomputer code instructs the computer to perform the harmonic expansionsuch that:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{ɛ_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}}}$where:  $ɛ_{0} = {\sum\limits_{k = 1}^{r}{n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}$

for the zeroth-order component, and$ɛ_{i} = {\sum\limits_{k = 1}^{r}{\frac{j\quad n_{k}^{2}}{\quad {{i2}\quad \pi}}{\quad\lbrack  { ( {{\cos ( {\frac{2\quad \pi \quad i}{D}x_{k - 1}} )} - {\quad{\quad{\cos ( {\frac{2\pi \quad i}{D}x_{k}} )}}}}  ) - {j( {{\sin ( {\frac{2\pi \quad i}{D}x_{k - 1}} )} - {\sin ( {\frac{2\quad \pi \quad i}{D}x_{k}} )}} )}} \rbrack }}}$

for the i^(th)-order harmonic component, where l indicates the lth oneof the plurality of hypothetical layers, D is the pitch of saidhypothetical deviated periodic structure, n_(k) is the index ofrefraction of a material between material boundaries at x_(k) andx_(k−1),j is the imaginary number defined as the square root of −1, andthere are r of said material boundaries within each period of saidhypothetical deviated periodic structure.
 39. A computer readablestorage medium containing computer executable code for generating anexpression of the permittivity of a target periodic grating having morethan two materials in a periodic direction for use in an opticalprofilometry formalism for determining a diffracted reflectivity of thetarget periodic grating by instructing a computer to operate as follows:divide the target periodic grating into a plurality of hypotheticallayers, at least one of the hypothetical layers formed across each of atleast a first, second and third material occurring along a line parallelto a direction of periodicity of the target periodic grating; subdivideat least one of the plurality of hypothetical layers into a plurality ofhypothetical slabs to generate a plurality of hypothetical boundaries,each of the plurality of hypothetical boundaries corresponding to anintersection of at least one of the plurality of hypothetical layerswith one of at least the first, second and third materials; determine apermittivity function for each of the plurality of hypothetical layers,where l indicates the lth one of the plurality of hypothetical layers;and complete a one-dimensional Fourier expansion of the permittivityfunction of each hypothetical layer along the direction of periodicityof the target periodic grating by summing the Fourier components overthe plurality of hypothetical boundaries to provide harmonic componentsof the at least one permittivity function. define a permittivityharmonics matrix including the harmonic components of the Fourierexpansion of the permittivity function.
 40. The computer readablestorage medium of claim 39 wherein in completing a one dimensionalFourier transform of a permittivity function the computer executablecode instructs the computer to complete the one dimensional Fouriertransform of the permittivity function ∈₁(x) according to:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{ɛ_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}}}$where  $ɛ_{0} = {\sum\limits_{k = 1}^{r}{n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}$

is the zeroth order component,$ɛ_{l,i} = {\sum\limits_{k = 1}^{r}{\frac{n_{k}^{2}}{{- {ji2}}\quad \pi}{\quad\lbrack  { ( {{\cos ( {\frac{2\quad i\quad \pi}{D}x_{k}} )} - {\quad{\quad{\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}}}}  ) - {j( {{\sin ( {\frac{2\quad i\quad \pi}{D}x_{k}} )} - {\sin ( {\frac{2\quad i\quad \pi}{D}x_{k - 1}} )}} )}} \rbrack }}}$

is the ith order component, l indicates the lth one of the plurality ofhypothetical layers, D is the pitch of the target periodic grating,n_(k) is the index of refraction of a material between each of theplurality of hypothetical boundaries at x_(k) and x_(k−1),j is theimaginary number defined as the square root of −1, and there are rhypothetical boundaries within each period of the target periodicgrating.
 41. The computer readable storage medium of claim 40 wherein indefining a permittivity harmonics matrix E_(l) the computer executablecode instructs the computer to generate a (2o+1)×(2o+1) Toeplitz-formmatrix having the form: $E_{l} = \begin{bmatrix}ɛ_{l,0} & ɛ_{l,{- 1}} & ɛ_{l,{- 2}} & \ldots & ɛ_{l,{{- 2}o}} \\ɛ_{l,1} & ɛ_{l,0} & ɛ_{l,{- 1}} & \ldots & ɛ_{l,{- {({{2o} - 1})}}} \\ɛ_{l,2} & ɛ_{l,1} & ɛ_{l,0} & \ldots & ɛ_{l,{- {({{2o} - 2})}}} \\\ldots & \ldots & \ldots & \ldots & \ldots \\ɛ_{l,{2o}} & ɛ_{l,{({{2o} - 1})}} & ɛ_{l,{({{2o} - 2})}} & \ldots & ɛ_{l,0}\end{bmatrix}$

where o is the order of the harmonic component.
 42. The computerreadable storage medium of claim 39 wherein in completing a onedimensional Fourier transform of a permittivity function the computerexecutable code instructs the computer to complete the one dimensionalFourier transform of the permittivity function π_(l)(x) according to:${\pi_{l}(x)} = {\frac{1}{ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{\pi_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}}}}$where  $\pi_{l,0} = {\sum\limits_{k = 1}^{r}{\frac{1}{n_{k}^{2}}\frac{x_{k} - x_{k - 1}}{D}}}$

is the zeroth order component,$\pi_{l,i} = {\sum\limits_{k = 1}^{r}{\frac{1}{\quad {{- {ji2}}\quad \pi}}\frac{1}{n_{k}^{2}}{\quad(  { ( {{\cos ( {\frac{2\quad \pi \quad i}{D}x_{k}} )} - {\quad{\quad{\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}}}}  ) - {j( {{\sin ( {\frac{2\pi \quad i}{D}x_{k}} )} - {\sin ( {\frac{2\quad \pi \quad i}{D}x_{k - 1}} )}} )}} ) }}}$

is the ith order component, l indicates the lth one of the plurality ofhypothetical layers, D is the pitch of the target periodic grating,n_(k) is the index of refraction of a material between each of theplurality of hypothetical boundaries at x_(k) and x_(k−1),j is theimaginary number defined as the square root of −1, and there are rhypothetical boundaries within each period of the target periodicgrating.
 43. The computer readable storage medium of claim 42 wherein indefining a permittivity harmonics matrix P_(l) the computer executablecode instructs the computer to generate a (2o+1)×(2o+1) Toeplitz-formmatrix having the form: $P_{l} = \begin{bmatrix}\pi_{l,0} & \pi_{l,{- 1}} & \pi_{l,{- 2}} & \ldots & \pi_{l,{{- 2}o}} \\\pi_{l,1} & \pi_{l,0} & \pi_{l,{- 1}} & \ldots & \pi_{l,{- {({{2o} - 1})}}} \\\pi_{l,2} & \pi_{l,1} & \pi_{l,0} & \ldots & \pi_{l,{- {({{2o} - 2})}}} \\\ldots & \ldots & \ldots & \ldots & \ldots \\\pi_{l,{2o}} & \pi_{l,{({{2o} - 1})}} & \pi_{l,{({{2o} - 2})}} & \ldots & \pi_{l,0}\end{bmatrix}$

where o is the order of the harmonic component.
 44. A system forgenerating a theoretical diffracted reflectivity associated withdiffraction of electromagnetic radiation off a target periodic gratingto determine structural properties of the target periodic grating,including a computer processor configured to: divide the target periodicgrating into a plurality of hypothetical layers, at least one of thehypothetical layers formed across each of at least a first, second andthird material, each of the at least first, second and third materialsoccurring along a direction of periodicity of the target periodicgrating, each separate hypothetical layer having one of a plurality ofpossible combinations of hypothetical values of properties for thathypothetical layer; generate sets of hypothetical layer data, each setof hypothetical layer data corresponding to a separate one of theplurality of hypothetical layers; and process the generated sets ofhypothetical layer data to generate the diffracted reflectivity thatwould occur by reflecting electromagnetic radiation off the periodicgrating.
 45. The system of claim 44 wherein the computer processor isfurther configured to subdivide the hypothetical layers into a pluralityof slabs, each slab corresponding to the intersection of one of theplurality of layers with one of at least the first, second and thirdmaterials.
 46. The system of claim 45 wherein in dividing the targetperiodic grating into a plurality of hypothetical layers the computerprocessor is further configured to divide the target periodic gratinginto a plurality of hypothetical layers which are parallel to thedirection of periodicity of the target periodic grating.
 47. The systemof claim 44 wherein in generating sets of hypothetical layer data thecomputer processor is configured to expand at least one of either afunction of a real space permittivity and a function of a real spaceinverse permittivity of the hypothetical layers in a one-dimensionalFourier transformation along the direction of periodicity of the targetperiodic grating to provide harmonic components of the at least one ofeither a function of a real space permittivity and a function of a realspace inverse permittivity of the hypothetical layers.
 48. The system ofclaim 44 wherein in generating sets of hypothetical layer data thecomputer processor is configured to compute at least one of:permittivity properties including a function of a permittivity ∈₁(x) ofeach of the hypothetical layers of the target periodic grating, theharmonic components ∈_(1,i) of the function of the permittivity ∈₁(x),and a permittivity harmonics matrix [E_(l)]; and inverse-permittivityproperties including a function of an inverse-permittivity π₁(x) of eachof the hypothetical layers of the target periodic grating, the harmoniccomponents π_(1,i) of the function of the inverse-permittivity π₁(x),and an inverse-permittivity harmonics matrix [P_(l)].
 49. The system ofclaim 48 wherein in processing the generated sets of hypothetical layerdata the computer processor is configured to: compute a wave-vectormatrix [A_(l)] by combining a series expansion of the electric field ofeach of the hypothetical layers of the target periodic grating with atleast one of at least the permittivity harmonics matrix [E_(l)] andinverse-permittivity harmonics matrix [P_(l)]. compute the ith entryw_(l,i,m) of the mth eigenvector of the wave-vector matrix [A_(l)] andthe mth eigenvalue τ_(l,m) of the wave-vector matrix [A_(l)] to form aneigenvector matrix [W_(l)] and a root-eigenvalue matrix [Q_(l)].
 50. Thesystem of claim 44 wherein in generating sets of hypothetical layer datathe computer processor is configured to expand one of at least afunction of a permittivity ∈₁(x) and a function of an inversepermittivity π₁(x)=1/∈₁(x) of the at least one of the hypotheticallayers formed across each of at least the first, second and thirdmaterials of the target periodic grating in a one-dimensional Fouriertransformation, the expansion performed along the direction ofperiodicity of the target periodic grating according to at least one of:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{ɛ_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}}}$where    $ɛ_{l,i} = {\sum\limits_{k = 1}^{r}{\frac{n_{k}^{2}}{{- {ji2}}\quad \pi}{\quad\lbrack ( {{{\cos ( {\frac{2i\quad \pi}{D}x_{k}} )} - { { {\quad{\quad{\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}}} ) - {j( {{\sin ( {\frac{2i\quad \pi}{D}x_{k}} )} - {\sin ( {\frac{2i\quad \pi}{D}x_{k - 1}} )}} )}} \rbrack {and}\quad {\pi_{l}(x)}}} = {\frac{1}{ɛ_{l}(x)} = {{\sum\limits_{i = {- \infty}}^{\infty}{\pi_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}{where}\quad \text{}\pi_{l,i}}} = {\sum\limits_{k = 1}^{r}{\frac{1}{{- {ji}}\quad 2\quad \pi}\frac{1}{n_{k}^{2}}( {( {{\cos ( {\frac{2\pi \quad i}{D}x_{k}} )} - {\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}} ) - {j( {{\sin ( {\frac{2\pi \quad i}{D}x_{k}} )} - {\sin ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}} )}} )}}}}}  }}}$

where l indicates the lth one of the plurality of hypothetical layers, Dis the pitch of said hypothetical deviated periodic structure, n_(k) isthe index of refraction of a material between material boundaries atx_(k) and X_(k−1),j is the imaginary number defined as the square rootof −1, and there are r of said material boundaries within each period ofsaid hypothetical deviated periodic structure.
 51. The system of claim44, wherein in processing the generated sets of hypothetical layer datathe computer processor is configured to: construct a matrix equationfrom the intermediate data corresponding to the hypothetical layers ofthe target periodic grating; and solve the constructed matrix equationto determine the diffracted reflectivity value R_(i) for each harmonicorder i.
 52. A system of generating the diffracted reflectivityassociated with diffraction of electromagnetic radiation off a targetperiodic grating to determine structural properties of the targetperiodic grating, including a computer microprocessor configured to:divide the target periodic grating into a plurality of hypotheticallayers, at least one of the hypothetical layers formed across each of atleast a first, second and third material occurring along a line parallelto a direction of periodicity of the target periodic grating; perform anharmonic expansion of a function of the permittivity ∈ along thedirection of periodicity of the target period grating for each of thehypothetical layers including the at least one of the plurality oflayers formed across each of at least a first, second and thirdmaterial; set up Fourier space electromagnetic equations in each of thehypothetical layers using the harmonic expansion of the function of thepermittivity ∈ for said each of the hypothetical layers and Fouriercomponents of electric and magnetic fields; couple the Fourier spaceelectromagnetic equations based on boundary conditions between thelayers; and solve the coupling of the Fourier space electromagneticequations to provide a diffracted reflectivity.
 53. The system of claim52 wherein the computer processor is further configured to subdivide atleast one of the plurality of hypothetical layers into a plurality ofhypothetical slabs, each hypothetical slab corresponding to anintersection of the at least one of the plurality of hypothetical layerswith one of at least the first, second and third materials.
 54. Thesystem of claim 53 wherein in subdividing at least one of thehypothetical layers into a plurality of hypothetical slabs, the computerprocessor is configured to subdivide the at least one hypothetical layerinto a plurality of hypothetical slabs such that only a single materiallies along any line perpendicular to the direction of periodicity of thetarget periodic grating and normal to the target periodic grating. 55.The system of claim 52 wherein in performing an harmonic expansion of afunction of the permittivity the computer processor is configured togenerate the harmonic expansion of the function of the permittivityalong the direction of periodicity of the target periodic grating forthe at least one of the hypothetical layers formed across each of atleast the first, second and third material such that:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{ɛ_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}}}$${where},\quad {ɛ_{0} = {\sum\limits_{k = 1}^{r}{n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}}$

for the zeroth-order component, and$ɛ_{i} = {\sum\limits_{k = 1}^{r}{\frac{j\quad n_{k}^{2}}{\quad {{i2}\quad \pi}}{\quad\lbrack  { ( {{\cos ( {\frac{2\quad \pi \quad i}{D}x_{k - 1}} )} - {\quad{\quad{\cos ( {\frac{2\pi \quad i}{D}x_{k}} )}}}}  ) - {j( {{\sin ( {\frac{2\pi \quad i}{D}x_{k - 1}} )} - {\sin ( {\frac{2\quad \pi \quad i}{D}x_{k}} )}} )}} \rbrack }}}$

for the i^(t)h-order harmonic component, and where l indicates the lthone of the plurality of hypothetical layers, D is the pitch of saidhypothetical deviated periodic structure, n_(k) is the index ofrefraction of a material between material boundaries at x_(k) andX_(k−1),j is the imaginary number defined as the square root of −1, andthere are r of said material boundaries within each period of saidhypothetical deviated periodic structure.
 56. A system of generating anexpression of the permittivity of a target periodic grating having morethan two materials in a periodic direction for use in an opticalprofilometry formalism for determining a diffracted reflectivity of thetarget periodic grating including a computer processor configured to:divide the target periodic grating into a plurality of hypotheticallayers, at least one of the hypothetical layers formed across each of atleast a first, second and third material occurring along a line parallelto a direction of periodicity of the target periodic grating; subdivideat least one of the plurality of hypothetical layers into a plurality ofhypothetical slabs to generate a plurality of hypothetical boundaries,each of the plurality of hypothetical boundaries corresponding to anintersection of at least one of the plurality of hypothetical layerswith one of at least the first, second and third materials; determine apermittivity function for each of the plurality of hypothetical layers;and complete a one-dimensional Fourier expansion of the permittivityfunction of each hypothetical layer along the direction of periodicityof the target periodic grating by summing the Fourier components overthe plurality of hypothetical boundaries to provide harmonic componentsof the at least one permittivity function. define a permittivityharmonics matrix including the harmonic components of the Fourierexpansion of the permittivity function.
 57. The system of claim 56wherein in completing a one dimensional Fourier transform of apermittivity function the computer processor is configured to complete aone dimensional Fourier transform of the permittivity function ∈₁(x)according to:${ɛ_{l}(x)} = {\sum\limits_{i = {- \infty}}^{\infty}{ɛ_{l,i}{\exp ( {j\frac{2\pi \quad i}{D}x} )}}}$where  $ɛ_{0} = {\sum\limits_{k = 1}^{r}{n_{k}^{2}\frac{x_{k - 1} - x_{k}}{D}}}$

is the zeroth order component,$ɛ_{l,i} = {\sum\limits_{k = 1}^{r}{\frac{n_{k}^{2}}{{- {ji2}}\quad \pi}{\quad\lbrack { ( {{\cos ( {\frac{2\quad i\quad \pi}{D}x_{k}} )} - {\quad{\quad{\cos ( {\frac{2\pi \quad i}{D}x_{k - 1}} )}}}}  ) - {j( {{\sin ( {\frac{2\quad i\quad \pi}{D}x_{k}} )} - {\sin ( {\frac{2\quad i\quad \pi}{D}x_{k - 1}} )}} )}} \rbrack}}}$

is the ith order component, l indicates the lth one of the plurality ofhypothetical layers, D is the pitch of the target periodic grating,n_(k) is the index of refraction of a material between each of theplurality of hypothetical boundaries at x_(k) and x_(k−1),j is theimaginary number defined as the square root of −1, and there are rhypothetical boundaries within each period of the target periodicgrating.
 58. The system of claim 57 wherein in defining the permittivityharmonics matrix E_(l) the computer processor constructs a (2o+1)×(2o+1)Toeplitz-form matrix having the form: $E_{l} = \begin{bmatrix}ɛ_{l,0} & ɛ_{l,{- 1}} & ɛ_{l,{- 2}} & \ldots & ɛ_{l,{{- 2}o}} \\ɛ_{l,1} & ɛ_{l,0} & ɛ_{l,{- 1}} & \ldots & ɛ_{l,{- {({{2o} - 1})}}} \\ɛ_{l,2} & ɛ_{l,1} & ɛ_{l,0} & \ldots & ɛ_{l,{- {({{2o} - 2})}}} \\\ldots & \ldots & \ldots & \ldots & \ldots \\ɛ_{l,{2o}} & ɛ_{l,{({{2o} - 1})}} & ɛ_{l,{({{2o} - 2})}} & \ldots & ɛ_{l,0}\end{bmatrix}$

where o is the order of the harmonic component.
 59. The system of claim56 wherein in completing a one dimensional Fourier transform of apermittivity function the computer processor completes a one dimensionalFourier transform of the permittivity function π_(l)(x) according to:${\pi_{l}(x)} = {\frac{1}{ɛ_{l}(x)} = {\underset{i = \infty}{\sum\limits^{\infty}}{\pi_{l,i}{\exp ( {j\frac{2\quad \pi \quad i}{D}x} )}}}}$where$\pi_{l,0} = {\underset{k = 1}{\sum\limits^{r}}{\frac{1}{n_{k}^{2}}\frac{x_{k} - x_{k - 1}}{D}}}$

is the zeroth order component,$\pi_{l,i} = {\underset{k = 1}{\sum\limits^{r}}{\frac{1}{{ji2}\quad \pi}\frac{1}{n_{k}^{2}}{\quad{\quad( {( {{\cos ( {\frac{2\quad \pi \quad i}{D}x_{k}} )} - {\cos ( {\frac{2\quad \pi \quad i}{D}x_{k - 1}} )}} ) - {j( {{\sin ( {\frac{2\quad \pi \quad i}{D}x_{k}} )} - {\sin ( {\frac{2\quad \pi \quad i}{D}x_{k - 1}} )}} )}} )}}}}$

is the ith order component, l indicates the lth one of the plurality ofhypothetical layers, D is the pitch of the target periodic grating,n_(k) is the index of refraction of a material between each of theplurality of hypothetical boundaries at x_(k) and X_(k−1),j is theimaginary number defined as the square root of −1, and there are rhypothetical boundaries within each period of the target periodicgrating.
 60. The system of claim 59 wherein in defining the permittivityharmonics matrix P_(l) the computer processor constructs a (2o+1)×(2o+1)Toeplitz-form matrix having the form: $P_{l} = \begin{bmatrix}\pi_{l,0} & \pi_{l,{- 1}} & \pi_{l,{- 2}} & \ldots & \pi_{l,{{- 2}o}} \\\pi_{l,1} & \pi_{l,0} & \pi_{l,{- 1}} & \ldots & \pi_{l,{- {({{2o} - 1})}}} \\\pi_{l{.2}} & \pi_{l,1} & \pi_{l{.0}} & \ldots & \pi_{l,{- {({{2o} - 2})}}} \\\ldots & \ldots & \ldots & \ldots & \ldots \\\pi_{l,{2o}} & \pi_{l,{({{2o} - 1})}} & \pi_{l,{({{2o} - 2})}} & \ldots & \pi_{l,0}\end{bmatrix}$

where o is the order of the harmonic component.